MTH 131

COURSE GUIDE

MTH 131

COURSE GUIDE

MTH 131 ELEMENTARY SET THEORY

Course Developer

Prof K. R. Adeboye eder!l U"#vers#$y of Te%&"olo'y M#"!.

Course (r#$er


Pro'r!))e *e!der

Dr. M!+!",uol! O+# S%&ool of S%#e"%e - Te%&"olo'y !$#o"!l Ope" U"#vers#$y U"#vers#$y of #'er#! *!'os

Course Co/ord#"!$or

0. Ab#ol!

!$#o"!l Ope" U"#vers#$y U"#vers#$y of #'er#! #'er#! *!'os

NATIONAL OPEN UNIVERSITY OF NIGERIA

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COURSE GUIDE

MTH 131

!$#o"!l Ope" U"#vers#$y U"#vers#$y of #'er#! He!du!r$ers 1214 A&)!du 0ello (!y 5#%$or#! Isl!"d *!'os Abu,! A""e6 728 S!)uel Adesu,o Ade)ule'u" S$ree$ Ce"$r!l 0us#"ess D#s$r#%$ Oppos#$e Are9! Su#$es Abu,! e/)!#l: %e"$r!l#"fo;"ou.edu."' UR*: 999."ou.edu."' !$#o"!l Ope" U"#vers#$y U"#vers#$y of #'er#! 7<<4 #rs$ Pr#"$ed 7<<4 IS0: =>?/<8?/727/? All R#'&$s Reserved Pr#"$ed by @@@@@.. or !$#o"!l Ope" U"#vers#$y U"#vers#$y of #'er#!

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COURSE GUIDE

MTH 131

Contents

Page

I"$rodu%$#o" .................................................. .................

1

(&!$ you 9#ll le!r" #" $&#s %ourse............................................. Course !#)s............................................................................... Course ob,e%$#ves....................................................................... (or+#"' $&rou'& $&#s %ourse...................................................... Course )!$er#!ls......................................................................... S$udy u"#$s................................................................................. Te6$ 0oo+................................................................................... Ass#'")e"$ #le ......................................................................... Ho9 $o 'e$ $&e )os$ fro) $&#s %ourse.........................................

1 1 7 7 7 7/3 3 3 3/2

#v

INTRODUCTION

(el%o)e $o Ele)e"$!ry se$ T&eory. T&#s %ourse #s ! 1/%red#$ u"#$ %ourse !"d #$ #s offered !$ $&e u"der'r!du!$e level. T&#s %ourse %o"s#s$s of e#'&$ ?B u"#$s T&ere !re "o %o)pulsory prereu#s#$es for $&#s %ourse T&#s Course Gu#de $ells you br#efly 9&!$ $&e %ourse #s !bou$ 9&!$ %ourse )!$er#!ls you 9#ll be us#"' !"d &o9 you %!" 9!l+ your 9!y $&rou'& $&ese )!$er#!ls. Wat Yo! W"## Lea$n In T"s Co!$se

T&e $&eory of se$s l#es !$ $&e fou"d!$#o"s of )!$&e)!$#%s. Co"%ep$s #" se$ $&eory su%& !s fu"%$#o"s !"d rel!$#o"s !ppe!r e6pl#%#$ly or #)pl#%#$ly #" every br!"%& of )!$&e)!$#%s. T&#s $e6$ #s !" #"for)!l "o"/!6#o)!$#% $re!$)e"$ of $&e $&eory of se$. T&e $e6$ %o"$!#"s !" #"$rodu%$#o" $o $&e ele)e"$!ry oper!$#o"s of se$s !"d ! de$!#led d#s%uss#o" of $&e %o"%ep$ of ! fu"%$#o" !"d ! rel!$#o". E!%& u"#$ be'#"s 9#$& %le!r s$!$e)e"$s of per$#"e"$ def#"#$#o"s pr#"%#ples !"d $&eore)s 9#ll #llus$r!$#ve !"d o$&er des%r#p$#ve )!$er#!ls. T&#s #s follo9ed by 'r!ded se$s of solved !"d supple)e"$!ry proble)s. T&e solved proble)s serve $o #llus$r!$e !"d !)pl#fy $&e $&eory br#"' #"$o s&!rp fo%us $&ose f#"e po#"$s 9#$&ou$ 9&#%& $&e s$ude"$ %o"$#"u!lly feels &#)self o" u"s!fe 'rou"d !"d prov#de $&e repe$#$#o" of b!s#% pr#"%#ples so v#$!l $o effe%$#ve le!r"#"'. Co!$se A"%s

T&e !#) of $&e %ourse %!" be su))!r#ed !s follo9s:/ • To #"$rodu%e you $o $&e b!s#% pr#"%#ples of se$ $&eory • (or+#"' 9#$& fu"%$#o"s !"d $&e "u)ber sys$e)

MTH 131

E*EMETAR SET THEOR

Co!$se O&'e(t")es

To !%&#eve $&e !#)s se$ ou$ !bove $&e %ourse se$s over!ll ob,e%$#ves. I" !dd#$#o" e!%& u"#$ !lso &!s spe%#f#% ob,e%$#ves. T&e u"#$ ob,e%$#ves !re !l9!ys #"%luded #" $&e be'#""#"' of ! u"#$F you s&ould re!d $&e) before you s$!r$ 9or+#"' $&rou'& $&e u"#$. ou )!y 9!"$ $o refer $o $&e) dur#"' your s$udy of $&e u"#$ $o %&e%+ o" your pro'ress. ou s&ould !l9!ys loo+ !$ $&e u"#$ ob,e%$#ves !f$er %o)ple$#"' ! u"#$. I" $&#s 9!y you %!" be sure $&!$ you &!ve do"e 9&!$ 9!s reu#red of you by $&e u"#$. Se$ ou$ belo9 !re $&e 9#der ob,e%$#ves of $&e %ourse !s ! 9&ole. 0y )ee$#"' $&ese ob,e%$#ves you s&ould &!ve !%&#eved $&e !#) of $&e %ourse !s ! 9&ole. O" su%%essful %o)ple$#o" of $&#s %ourse you s&ould be !ble $o:/ • • • •

E6pl!#" se$s subse$s se$ "o$!$#o"s +#"ds of se$s Des%r#be b!s#% se$ oper!$#o"s Ide"$#fy #"$erv!ls !s se$s !"d des%r#be #"$erv!ls us#"' $&e re!l l#"e Ide"$#fy fu"%$#o"s !"d $&e rel!$#o"s&#p be$9ee" produ%$ se$s !"d 'r!p&s of fu"%$#o"s • Apply $&e l!9s of $&e !l'ebr! of se$s #" prov#"' se$ #de"$#$#es Wo$*"ng t$o!g t"s Co!$se

To %o)ple$e $&#s %ourse you !re reu#red $o re!d $&e s$udy u"#$s re!d se$ boo+s !"d re!d o$&er )!$er#!ls prov#ded by $&e OU Co!$se Mate$"a#s

M!,or %o)po"e"$s of $&e %ourse !re:/ 1. Course Gu#de 7. S$udy U"#$s 3. Te6$ boo+s 2. Ass#'")e"$ #le St!+, Un"ts

T&ere !re e#'&$ s$udy u"#$s #" $&#s %ourse !s follo9s:/ U"#$ 1: Se$ !"d Subse$s U"#$ 7: 0!s#% Se$ Oper!$#o"s U"#$ 3: Se$s of u)bers U"#$ 2: u"%$#o"s ##

MTH 131


U"#$ 8: Produ%$ Se$s !"d Gr!p&s of u"%$#o"s U"#$ 4: Rel!$#o"s U"#$ >: ur$&er T&eory of Se$s U"#$ ?: ur$&er T&eory of u"%$#o"s !"d Oper!$#o"s Te-t oo*s

T&ere !re "o %o)pulsory $e6$ boo+s for $&#s %ourse Ass"gn%ent F"#e

T&e !ss#'")e"$ #le %o"$!#"s de$!#ls of $&e 9or+ you )us$ sub)#$ $o your $u$or for )!r+#"'. I$ %o"$!#"s ! )ore %o)p!%$ for) of $&e Tu$or/)!r+ed !ss#'")e"$s. T&ere !re ! )!6#)u) of f#ve !ss#'")e"$s #" e!%& u"#$ Assess%ent

T&ere !re $9o !spe%$s of $&e !ssess)e"$ of $&e %ourse. #rs$ !re $&e $u$or/ )!r+ed !ss#'")e"$sF se%o"d $&ere #s ! 9r#$$e" e6!)#"!$#o". I" $!%+l#"' $&e !ss#'")e"$s you !re e6pe%$ed $o !pply #"for)!$#o" +"o9led'e !"d $e%&"#ues '!$&ered dur#"' $&e %ourse. T&e !ss#'")e"$s )us$ be sub)#$$ed $o your $u$or for for)!l !ssess)e"$ #" !%%ord!"%e 9#$& $&e s$#pul!$ed de!dl#"es. Ho/ to get te %ost 0$o% te (o!$se

I" d#s$!"%e le!r"#"' $&e s$udy u"#$s repl!%e $&e le%$urer. T&#s #s o"e of $&e 're!$ !dv!"$!'es of d#s$!"%e le!r"#"'F you %!" re!d !"d 9or+ $&rou'& spe%#!lly des#'"ed s$udy )!$er#!ls !$ your p!%e !"d !$ ! $#)e !"d pl!%e $&!$ su#$ you bes$. T&#"+ of #$ !s re!d#"' $&e le%$ure #"s$e!d of l#s$e"#"' $o ! le%$urer. I" $&e s!)e 9!y $&!$ ! le%$urer )#'&$ se$ you so)e re!d#"' $o do $&e s$udy u"#$s $ell you 9&e" $o re!d your se$ boo+s or o$&er )!$er#!ls !"d 9&e" $o u"der$!+e %o)pu$#"' pr!%$#%!l 9or+. us$ !s ! le%$urer )#'&$ '#ve you !" #"/%l!ss e6er%#se your s$udy u"#$s prov#de e6er%#ses for you $o do !$ !ppropr#!$e po#"$s. E!%& of $&e s$udy u"#$s follo9s ! %o))o" for)!$. T&e f#rs$ #$e) #s !" #"$rodu%$#o" $o $&e sub,e%$ )!$$er of $&e u"#$ !"d &o9 ! p!r$#%ul!r u"#$ #s #"$e'r!$ed 9#$& $&e o$&er u"#$s !"d $&e %ourse !s ! 9&ole. e6$ #s ! se$ of le!r"#"' ob,e%$#ves. T&ese ob,e%$#ves le$ you +"o9 9&!$ you s&ould be !ble $o do by $&e $#)e you &!ve %o)ple$ed $&e u"#$. ou s&ould use $&ese ob,e%$#ves $o 'u#de your s$udy. (&e" you &!ve f#"#s&ed $&e u"#$ you )us$ 'o ###

MTH 131


b!%+ !"d %&e%+ 9&e$&er you &!ve !%&#eved $&e ob,e%$#ves. If you )!+e ! &!b#$ of do#"' $&#s you 9#ll s#'"#f#%!"$ly #)prove your %&!"%es of p!ss#"' $&e %ourse. E6er%#ses !re #"$erspersed 9#$&#" $&e u"#$s !"d !"s9ers !re '#ve". (or+#"' $&rou'& $&ese e6er%#ses 9#ll &elp you $o !%&#eve $&e ob,e%$#ves of $&e u"#$ !"d &elp you $o prep!re for $&e !ss#'")e"$s !"d e6!)#"!$#o". T&e follo9#"' #s ! pr!%$#%!l s$r!$e'y for 9or+#"' $&rou'& $&e %ourse. 1.

Re!d $&#s Course Gu#de $&orou'&ly

7.

Or'!"#e ! s$udy s%&edule. Refer $o $&e Course Overv#e9 for )ore de$!#ls

3.

O"%e you &!ve %re!$ed your o9" s$udy s%&edule do every$&#"' you %!" $o s$#%+ $o #$. T&e )!,or re!so" $&!$ s$ude"$s f!#ls #s $&!$ $&ey 'e$ be&#"d 9#$& $&e#r %ourse 9or+. If you 'e$ #"$o d#ff#%ul$#es 9#$& your s%&edule ple!se le$ your $u$or +"o9 before #$ #s $oo l!$e.

2.

Tur" $o U"#$ 1 !"d re!d $&e #"$rodu%$#o" !"d $&e ob,e%$#ves for $&e u"#$.

8.

(or+ $&rou'& $&e u"#$. T&e %o"$e"$ of $&e u"#$ #$self &!s bee" !rr!"'ed $o prov#de ! seue"%e for you $o follo9. Rev#e9 $&e ob,e%$#ves for e!%& s$udy u"#$ $o %o"f#r) $&!$ you &!ve !%&#eved $&e). If you feel u"sure !bou$ !"y of $&e ob,e%$#ves rev#e9 $&e s$udy )!$er#!ls or %o"sul$ your $u$or.

4.

#v

>.

(&e" you !re %o"f#de"$ $&!$ you &!ve !%&#eved ! u"#$Js ob,e%$#ves you %!" $&e" s$!r$ o" $&e "e6$ u"#$. Pro%eed u"#$ by u"#$ $&rou'& $&e %ourse !"d $ry $o p!%e your s$udy so $&!$ you +eep yourself o" s%&edule.

?.

(&e" you &!ve sub)#$$ed !" !ss#'")e"$ $o your $u$or for )!r+#"' do "o$ 9!#$ for #$s re$ur" before s$!r$#"' o" $&e "e6$ u"#$. Keep $o your s%&edule. (&e" $&e !ss#'")e"$ #s re$ur"ed p!y p!r$#%ul!r !$$e"$#o" $o your $u$orJs %o))e"$s.

=.

Af$er %o)ple$#"' $&e l!s$ u"#$ rev#e9 $&e %ourse !"d prep!re yourself for f#"!l e6!)#"!$#o". C&e%+ $&!$ you &!ve !%&#eved $&e u"#$ ob,e%$#ves l#s$ed !$ $&e be'#""#"' of e!%& u"#$B !"d $&e %ourse ob,e%$#ves l#s$ed o" $&#s Course Gu#de.

MTH 131


MAIN COURSE

Course Code

MTH 131

Course T#$le

E*EMETAR SET THOER

Course Developer


Course (r#$er


Pro'r!))e *e!der

Dr. M!+!",uol! O+# S%&ool of S%#e"%e - Te%&"olo'y !$#o"!l Ope" U"#vers#$y of #'er#! *!'os

Course Co/ord#"!$or

0. Ab#ol!

!$#o"!l Ope" U"#vers#$y of #'er#! *!'os

NATIONAL OPEN UNIVERSITY OF NIGERIA

v

MTH 131


He!du!r$ers 1214 A&)!du 0ello (!y 5#%$or#! Isl!"d *!'os Abu,! A""e6 728 S!)uel Adesu,o Ade)ule'u" S$ree$ Ce"$r!l 0us#"ess D#s$r#%$ Oppos#$e Are9! Su#$es Abu,! e/)!#l: %e"$r!l#"fo;"ou.edu."' UR*: 999."ou.edu."' !$#o"!l Ope" U"#vers#$y of #'er#! 7<<4 #rs$ Pr#"$ed 7<<4 IS0: =>?/<8?/727/? All R#'&$s Reserved Pr#"$ed by @@@@@.. or !$#o"!l Ope" U"#vers#$y of #'er#!

Ta&#e o0 Content

Mo+!#e 1

v#

Page

MTH 131

U"#$ 1 U"#$ 7 U"#$ 3 U"#$ 2 U"#$ 8


Se$s - Subse$s......................................... 1/18 0!s#% Se$ Oper!$#o"s............................ 14/74 Se$ of u)bers....................................... 7>/3? u"%$#o"s................................................. 3=/88 u"%$#o"s............................................... 84/4?

Mo+!#e 

U"#$ 1 U"#$ 7 U"#$ 3

Rel!$#o"s................................................ 4=/>4 ur$&er T&eory of Se$s.......................... >>/?> ur$&er T&eory of u"%$#o"s Oper!$#o" ............................................... ??/1<2

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MTH 131


Mo+!#e 1

U"#$ 1 U"#$ 7 U"#$ 3 U"#$ 2 U"#$ 8

Se$s - Subse$s 0!s#% Se$ Oper!$#o"s Se$ of u)bers u"%$#o"s u"%$#o"s

UNIT 1

SETS 2 SUSETS

CONTENTS

1.< 7.< 3.<

2.< 8.< 4.< >.<

I"$rodu%$#o" Ob,e%$#ves M!#" 0ody 3.1 Se$s 3.1.1 o$!$#o" 3.1.7 #"#$e !"d I"f#"#$e se$s 3.1.3 Eu!l#$y of se$s 3.1.3 ull Se$ 3.7 Subse$s 3.7.1 Proper subse$s 3.7.7 Co)p!r!b#l#$y 3.7.3 Se$ of se$s 3.7.2 U"#vers!l se$ 3.7.8 Po9er se$ 3.7.4 D#s,o#"$ se$s 3.3 5e""/Euler d#!'r!)s 3.2 A6#o)!$#% develop)e"$ of se$ $&eory Co"%lus#o" Su))!ry Tu$or/M!r+ed !ss#'")e"$ Refere"%es !"d ur$&er re!d#"'s

1

MTH 131

1.


INTRODUCTION

T&e $&eory of se$s l#es !$ $&e fou"d!$#o" of )!$&e)!$#%s. I$ #s ! %o"%ep$ $&!$ re!rs #$s &e!d #" !l)os$ !ll f#elds of )!$&e)!$#%sF pure !"d !ppl#ed. T&#s u"#$ !#)s !$ #"$rodu%#"' b!s#% %o"%ep$s $&!$ 9ould be e6pl!#"ed ur$&er #" subseue"$ u"#$s. T&ere 9#ll be def#"#$#o" of $er)s !"d lo$s of e6!)ples !"d e6er%#ses $o &elp you !s you 'o !lo"'. .

O4ECTIVES

A$ $&e e"d of $&#s u"#$ you s&ould be !ble $o: • Ide"$#fy se$s fro) so)e '#ve" s$!$e)e"$s • Re9r#$e se$s #" $&e d#ffere"$ se$ "o$!$#o" • Ide"$#fy $&e d#ffere"$ +#"ds of se$s 9#$& e6!)ples 3.

MAIN ODY

3.1

Sets

As )e"$#o"ed #" $&e #"$rodu%$#o" ! fu"d!)e"$!l %o"%ep$ #" !ll ! br!"%& of )!$&e)!$#%s #s $&!$ of se$. Here #s ! def#"#$#o"  A set is any well-defined list, collection or class of objects”. T&e ob,e%$s #" se$s !s 9e s&!ll see fro) e6!)ples %!" be !"y$&#"': 0u$ for %l!r#$y 9e "o9 l#s$ $e" p!r$#%ul!r e6!)ples of se$s: E-a%5#e 1.1 E-a%5#e 1. E-a%5#e 1.3 E-a%5#e 1.6 E-a%5#e 1.7 E-a%5#e 1.8 E-a%5#e 1.9 E-a%5#e 1.: E-a%5#e 1.; E-a%5#e 1.1

7

T&e "u)bers <724? T&e solu$#o"s of $&e eu!$#o" 6 7  761 L < T&e vo9els of $&e !lp&!be$: ! e # o u T&e people l#v#"' o" e!r$& T&e s$ude"$s To) D#%+ !"d H!rry T&e s$ude"$s 9&o !re !bse"$ fro) s%&ool T&e %ou"$r#es E"'l!"d r!"%e !"d De")!r+ T&e %!p#$!l %#$#es of #'er#! T&e "u)ber 1 3 > !"d 1< T&e r#vers #" #'er#!

MTH 131


o$e $&!$ $&e se$s #" $&e odd "u)bered e6!)ples !re def#"ed $&!$ #s prese"$ed by !%$u!lly l#s$#"' #$s )e)bersF !"d $&e se$s #" $&e eve" "u)bered e6!)ples !re def#"ed by s$!$#"' proper$#es $&!$ #s rules 9&#%& de%#de 9&e$&er or "o$ ! p!r$#%ul!r ob,e%$ #s ! )e)ber of $&e se$. 3.1.1 Notat"on

Se$s 9#ll usu!lly be de"o$ed by %!p#$!l le$$ersF A 0  @@ *o9er %!se le$$ers 9#ll usu!lly represe"$ $&e ele)e"$s #" our se$s: *e$s $!+e !s !" e6!)pleF #f 9e def#"e ! p!r$#%ul!r se$ by !%$u!lly l#s$#"' #$s )e)bers for e6!)ple le$ A %o"s#s$ of "u)bers 13> !"d 1< $&e" 9e 9r#$e ALN13>1< T&!$ #s $&e ele)e"$s !re sep!r!$ed by %o))!s !"d e"%losed #" br!%+e$s N. (e %!ll $&#s $&e ta&!#a$ 0o$% of ! se$ o9 $ry your &!"d o" $&#s E-e$("se 1.1

1. 7. 3. 2.

S$!$e #" 9ords !"d $&e" 9r#$e #" $!bul!r for) A L N6 Q67 L 27 0 L N6Q6  7 L 8 C L N6 Q6 #s pos#$#ve 6 #s "e'!$#ve D L N6 Q6 #s ! le$$er #" $&e 9ord %orre%$

So#!t"on<

1. 7. 3.

I$ re!ds A #s $&e se$ of 6 su%& $&!$ 6 su!red eu!ls four. T&e o"ly "u)bers 9&#%& 9&e" su!red '#ve four !re 7 !"d /7. He"%e A L N7 /7 I$ re!ds 0 #s $&e se$ of 6 su%& $&!$ 6 )#"us 7 eu!ls 8. T&e o"ly solu$#o" #s >F &e"%e 0 L N> I$ re!d C #s $&e se$ of 6 su%& $&!$ 6 #s pos#$#ve !"d 6 #s "e'!$#ve. T&ere #s "o "u)ber 9&#%& #s bo$& pos#$#ve !"d "e'!$#veF &e"%e C #s e)p$y $&!$ #s CL 3

MTH 131

2.


I$ re!ds D #s $&e se$ of 6 su%& $&!$ 6 #s le$$er #" $&e 9or+ %orre%$J. T&e #"d#%!$ed le$$ers !re %ore !"d $F $&us D L N%ore$

0u$ #f 9e def#"e ! p!r$#%ul!r se$ by s$!$#"' proper$#es 9&#%& #$s ele)e"$s )us$ s!$#sfy for e6!)ple le$ 0 be $&e se$ of !ll eve" "u)bers $&e" 9e use ! le$$er usu!lly 6 $o represe"$ !" !rb#$r!ry ele)e"$ !"d 9e 9r#$e: 0 L N6Q6 #s eve" (&#%& re!ds 0 #s $&e se$ of "u)bers 6 su%& $&!$ 6 #s eve". (e %!ll $&#s $&e se$ &!"#+e$s 0o$% of ! se$. o$#%e $&!$ $&e ver$#%!l l#"e Q #s re!d su%& !s. I" order $o #llus$r!$e $&e use of &$ !bove "o$!$#o"s 9e re9r#$e $&e se$s #" e6!)ples 1.1/1.1<. (e de"o$e $&e se$s by A 1 A7 @.. A1< respe%$#vely. E-a%5#e .1< E-a%5#e .< E-a%5#e .3< E-a%5#e .6< E-a%5#e .7< E-a%5#e .8< E-a%5#e .9< E-a%5#e .:< E-a%5#e .;< E-a%5#e .1<

A1 L N< 7 2 4 ? A7 L N6Q67 76  1 L < A3 L N! e # o u A2 L N6 Q6 #s ! perso" l#v#"' o" $&e e!r$& A8 L NTo) D#%+ H!rry A4 L N6 Q 6 #s ! s$ude"$ !"d 6 #s !bse"$ fro) s%&ool A> L NE"'l!"d r!"%e De")!r+ A? L N6Q6 #s ! %!p#$!l %#$y !"d 6 #s #" #'er#! A= L N1 3 > 1< A1< L N 6Q6 #s ! r#ver !"d 6 #s #" #'er#!

I$ #s e!sy !s $&!$ E-e$("se 1. W$"te Tese Sets In A Set=!"#+e$ Fo

%$1. 7. 3. 2. 8.

2

*e$ A %o"s#s$ of $&e le$$ers ! b % d !"d e *e$ 0 L N7 2 4 ?@@.. *e$ C %o"s#s$ of $&e %ou"$r#es #" $&e U"#$ed !$#o"s *e$ D L N3 *e$ E be $&e He!ds of S$!$e 0!b!"'#d! Ab!%&! !"d Abduls!l!)#

MTH 131


So#!t"on

1. 7. 3. 2. 8.

A L N6 Q6 !ppe!rs before f #" $&e !lp&!be$ L N6 Q6 #s o"e of $&e f#rs$ le$$ers #" $&e !lp&!be$ 0 L N6 Q6 #s eve" !"d pos#$#ve C L N6 Q6 #s ! %ou"$ry 6 #s #" $&e U"#$ed !$#o"s D L N6 Q6  7 L 1 L N6Q76 L 4 E L N6 Q6 9!s He!d of s$!$e !f$er 0u&!r#

If !" ob,e%$ 6 #s ! )e)ber of ! se$ A #.e. A %o"$!#"s 6 !s o"e of #$sV ele)e"$s $&e" 9e 9r#$e: 6 ∈A 9&#%& %!" be re!d 6 belo"'s $o A or 6 #s #" A. If o" $&e o$&er &!"d !" ob,e%$ 6 #s "o$ ! )e)ber of ! se$ A #.e A does "o$ %o"$!#" 6 !s o"e of #$s ele)e"$s $&e" 9e 9r#$eF 6 ∉A I$ #s ! %o))o" %us$o) #" )!$&e)!$#%s $o pu$ ! ver$#%!l l#"e    or \  $&rou'& ! sy)bol $o #"d#%!$e $&e oppos#$e or "e'!$#ve )e!"#"' of $&e sy)bol. E-a%5#e 3<1< E-a%5#e 3.<

3.1.1

*e$ A L N! e # o u. T&e" !∈A b∉A f ∉A. *e$ 0 L N6 6 #s eve". T&e" 3∉0 4∈0 11∉0 12 ∈ 0.

F"n"te 2 In0"n"te Sets

Se$s %!" be f#"#$e or #"f#"#$e. I"$u#$#vely ! se$ #s f#"#$e #f #$ %o"s#s$s of ! s5e("0"( n!%&e$ of d#ffere"$ ele)e"$s #.e. #f #" %ou"$#"' $&e d#ffere"$ )e)bers of $&e se$ $&e %ou"$#"' pro%ess %!" %o)e $o !" e"d. O$&er9#se ! se$ #s #"f#"#$e. *e$s loo+ !$ so)e e6!)ples. E-a%5#e 6<1<

*e$ M be $&e se$ of $&e d!ys of $&e 9ee+. T&e M #s f#"#$e

E-a%5#e 6<<

*e$  L N<724?@@... T&e"  #s #"f#"#$e

8

MTH 131


E-a%5#e 6<3<

*e$ P L N6 6 #s ! r#ver o" $&e e!r$&. Al$&ou'& #$ )!y be d#ff#%ul$ $o %ou"$ $&e "u)ber of r#vers #" $&e 9orld P #s s$#ll ! f#"#$e se$.

E-e$("se 1.3<

9&#%& se$s !re f#"#$eW

1. 7. 3. 2. 8.

T&e )o"$&s of $&e ye!r N1 7 3 @@@ == 1<< T&e people l#v#"' o" $&e e!r$& N6 X 6 #s eve" N1 7 3@@..

So#!t"on<

T&e f#rs$ $&ree se$s !re f#"#$e. Al$&ou'& p&ys#%!lly #$ )#'&$ be #)poss#ble $o %ou"$ $&e "u)ber of people o" $&e e!r$& $&e se$ #s s$#ll f#"#$e. T&e l!s$ $9o se$s !re #"f#"#$e. If 9e ever $ry $o %ou"$ $&e eve" "u)bers 9e 9ould "ever %o)e $o $&e e"d. 3.1. E>!a#"t, O0 Sets

Se$ A #s e>!a# $o se$ 0 #f $&ey bo$& &!ve $&e s!)e )e)bers #.e #f every ele)e"$ 9&#%& belo"'s $o A !lso belo"'s $o 0 !"d #f every ele)e"$ 9&#%& belo"'s $o 0 !lso belo"'s $o A. (e de"o$e $&e eu!l#$y of se$s A !"d 0 by: AL0 E-a%5#e 7.1

*e$ A L N1 7 3 2 !"d 0 L N3 1 2 7. T&e" A L 0 $&!$ #s N1732 L N3127 s#"%e e!%& of $&e ele)e"$s 173 !"d 2 of A belo"'s $o 0 !"d e!%& of $&e ele)e"$s 312 !"d 7 of 0 belo"'s $o A. o$e $&erefore $&!$ ! se$ does "o$ %&!"'e #f #$s ele)e"$s !re re!rr!"'ed.

E-a%5#e 7.3

*e$ ELN6 X 67 36 L /7 LN71 !"d G LN177 1 T&e" EL L G

4

MTH 131


3.1.3 N!## Set

I$ #s %o"ve"#e"$ $o #"$rodu%e $&e %o"%ep$ of $&e e)p$y se$ $&!$ #s ! se$ 9&#%& %o"$!#"s "o ele)e"$s. T&#s se$ #s so)e$#)es %!lled $&e null set. (e s!y $&!$ su%& ! se$ #s vo#d or e)p$y !"d 9e de"o$e #$s sy)bol ∅. E-a%5#e 8.1 *e$ A be $&e se$ of people #" $&e 9orld 9&o !re older $&!"

7<< ye!rs. A%%ord#"' $o +"o9" s$!$#s$#%s A #s $&e "ull se$.

E-a%5#e 8. *e$ 0 L N6 X 67 L 2 6 #s odd T&e" 0 #s $&e e)p$y se$.

3.7

S!&sets

If every ele)e"$ #" ! se$ A #s !lso ! )e)ber of ! se$ 0 $&e" A #s %!lled subset of 0. More spe%#f#%!lly A #s ! subse$ of 0 #f 6 ∈A #)pl#es 6 ∈0. (e de"o$e $&#s rel!$#o"s&#p by 9r#$#"'F A ⊂ 0 9&#%& %!" !lso be re!d A #s %o"$!#"ed #" 0. E-a%5#e 9.1 T&e

se$ C L N138 #s ! subse$ of D L N82371 s#"%e e!%& "u)ber 1 3 !"d 8 belo"'#"' $o C !lso belo"'s $o D.

E-a%5#e 9. T&e se$ E L N724 #s ! subse$ of  L N472 s#"%e e!%&

"u)ber 72 !"d 4 belo"'#"' $o E !lso belo"'s $o . o$e #" p!r$#%ul!r $&!$ E L . I" ! s#)#l!r )!""er #$ %!" be s&o9" $&!$ every se$ #s ! subse$ of #$self.

E-a%5#e 9.3 *e$ G L N6 X 6 #s eve" #.e. G L N724 !"d le$  L N6 X 6

#s ! pos#$#ve po9er of 7 #.e. le$  L N72?14@.. T&e"  ⊂ G #.e.  #s %o"$!#"ed #" G.

(#$& $&e !bove def#"#$#o" of ! subse$ 9e !re !ble $o res$!$e $&e def#"#$#o" of $&e eu!l#$y of $9o se$s. T9o se$ A !"d 0 !re eu!l #.e A L 0 #f !" o"ly #f A ⊂ 0 !"d 0⊂A. If A #s ! subse$ of 0 $&e" 9e %!" !lso 9r#$e 0 ⊃A >

MTH 131


9&#%& re!ds 0 #s ! superse$ of A or 0 %o"$!#"s A. ur$&er)ore 9e 9r#$e: A ⊄ 0 #f A #s "o$ ! subse$ of 0. Co"%lus#vely 9e s$!$e: 1. 7.

T&e "ull se$ ∅ #s %o"s#dered $o be ! subse$ of every se$ If A #s "o$ ! subse$ of 0 $&!$ #s #f A ⊄ 0 $&e" $&ere #s !$ le!s$ o"e ele)e"$ #" A $&!$ #s "o$ ! )e)ber of 0.

3..1 P$o5e$ S!&sets

S#"%e every se$ A #s ! subse$ of #$self 9e %!ll 0 ! proper subse$ of A #f f#rs$ #s ! subse$ of A !"d se%o"dly #f 0 #s "o$ eu!l $o A. More br#efly 0 #s ! proper subse$ of A #f: 0 ⊂ A !"d 0 ≠ A I" so)e boo+s 0 #s ! subse$ of A #s de"o$ed by 0 ⊆ A !"d 0 #s ! proper subse$ of A #s de"o$ed by 0 ⊂ A (e 9#ll %o"$#"ue $o use $&e prev#ous "o$!$#o" #" 9&#%& 9e do "o$ d#s$#"'u#s&ed be$9ee" ! subse$ !"d ! proper subse$. 3.. Co%5a$a&"#"t,

T9o se$s A !"d 0 !re s!#d $o be (o%5a$a&#e #f: A ⊂ 0 or 0 ⊂ AF T&!$ #s #f o"e of $&e se$s #s ! subse$ of $&e o$&er se$. Moreover $9o se$s A !"d 0 !re s!#d $o be not (o%5a$a&#e #f: A ⊄ 0 !"d 0 ⊄ A ?

MTH 131


o$e $&!$ #f A #s "o$ %o)p!r!ble $o 0 $&e" $&ere #s !" ele)e"$ #" A .9&#%& #s "o$ #" 0 !"d @ !lso $&ere #s !" ele)e"$ #" 0 9&#%& #s "o$ #" A. E-a%5#e :.1<

*e$ A L N!b !"d 0 L N!b%. T&e A #s %o)p!r!ble $o 0 s#"%e A #s ! subse$ of 0.

E-a%5#e :.<

*e$ R  N!b !"d S L Nb%d. T&e" R !"d S !re "o$ %o)p!r!ble s#"%e ! ∈R "d !∉ S !"d % ∉ R

I" )!$&e)!$#%s )!"y s$!$e)e"$s %!" be prove" $o be $rue by $&e use of prev#ous !ssu)p$#o"s !"d def#"#$#o"s. I" f!%$ $&e esse"%e of )!$&e)!$#%s %o"s#s$s of $&eore)s !"d $&e#r proofs. (e "o9 proof our f#rs$ Teo$e% 1.1 If

A #s ! subse$ of 0 !"d 0 #s ! subse$ of C $&e" A #s ! subse$ of C $&!$ #s A ⊂ 0 !"d 0 ⊂ C #)pl#es A ⊂ 0 o$#%e $&!$ 9e )us$ s&o9 $&!$ !"y ele)e"$ #" A #s !lso !" ele)e"$ #" CB. *e$ 6 be !" ele)e"$ of A $&!$ #s le$ 6 ∈ A. S#"%e A #s ! subse$ of 0 6 !lso belo"'s $o 0 $&!$ #s 6 ∈ 0. 0u$ by &ypo$&es#s 0 ⊂ CF &e"%e every ele)e"$ of 0 9&#%& #"%ludes 6 #s ! "u)ber of C. (e &!ve s&o9" $&!$ 6 ∈ A #)pl#es 6 ∈ C. A%%ord#"'ly by def#"#$#o" A ⊂ C. P$oo0<

3..3 Sets o0 Sets

I$ so)e$#)es 9#ll &!ppe" $&!$ $&e ob,e%$ of ! se$ !re se$s $&e)selvesF for e6!)ple $&e se$ of !ll subse$s of A. I" order $o !vo#d s!y#"' se$ of se$s #$ #s %o))o" pr!%$#%e $o s!y f!)#ly of se$s or %l!ss of se$s. U"der $&e %#r%u)s$!"%es !"d #" order $o !vo#d %o"fus#o" 9e so)e$#)es 9#ll le$ s%r#p$ le$$ers

De"o$e f!)#l#es or %l!sses of se$s s#"%e %!p#$!l le$$ers !lre!dy de"o$e $&e#r ele)e"$s. =

MTH 131


E-a%5#e ;.1:

I" 'eo)e$ry 9e usu!lly s!y ! f!)#ly of l#"es or ! f!)#ly of %urves s#"%e l#"es !"d %urves !re $&e)selves se$s of po#"$s.

E-a%5#e ;.<

T&e se$ NN73 N7 N84 #s ! f!)#ly of se$s. I$s )e)bers !re $&e se$s N73 N7 !"d N84.

T&eore$#%!lly #$ #s poss#ble $&!$ ! se$ &!s so)e )e)bers 9&#%& !re se$s $&e)selves !"d so)e )e)bers 9&#%& !re "o$ se$s !l$&ou'& #" !"y !ppl#%!$#o" of $&e $&eory of se$s $&#s %!se !r#ses #"freue"$ly. E-a%5#e ;.3

*e$ A L N7 N13 2 N78. T&e" A #s "o$ ! f!)#ly of se$sF &ere so)e ele)e"$s of A !re se$s !"d so)e !re "o$.

3..6 Un")e$sa# Set

I" !"y !ppl#%!$#o" of $&e $&eory of se$s !ll $&e se$s u"der #"ves$#'!$#o" 9#ll l#+ely be subse$s of ! f#6ed se$. (e %!ll $&#s se$ $&e universal set or universe of discourse. (e de"o$e $&#s se$ by U. E-a%5#e 1.1< I"

pl!"e 'eo)e$ry $&e u"#vers!l se$ %o"s#s$s of !ll $&e po#"$s #" $&e pl!"e. E-a%5#e 1.

I" &u)!" popul!$#o" s$ud#es $&e u"#vers!l se$ %o"s#s$s of !ll $&e people #" $&e 9orld.

3..7 Po/e$ Set

T&e f!)#ly of !ll $&e subse$s of !"y se$ S #s %!lled $&e 5o/e$ set of S. (e de"o$e $&e po9er se$ of S by: 7 S E-a%5#e11.1*e$

M L N!b T&e" 7M L NN! b N! Nb ∅

E-a%5#e 11.

*e$ T L N2>? $&e" 7T L NT N2> N2? N>? N2 N> N? ∅

If ! se$ S #s f#"#$e s!y S &!s " ele)e"$s $&e" $&e po9er se$ of S %!" be s&o9" $o &!ve 7" ele)e"$s. T&#s #s o"e re!so" 9&y $&e %l!ss of subse$s of S #s %!lled $&e po9er se$ of S !"d #s de"o$ed by 7 S 1<

MTH 131


3..8 D"s'o"nt Sets

If se$s A !"d 0 &!ve "o ele)e"$s #" %o))o" #.e #f "o ele)e"$ of A #s #" 0 !"d "o ele)e"$ of 0 #s #" A $&e" 9e s!y $&!$ A !"d 0 !re +"s'o"nt E-a%5#e 1.1<

*e$ A L N13>? !"d 0 L N72>= T&e" A !"d 0 !re "o$ d#s,o#"$ s#"%e > #s #" bo$& se$s #.e > ∈ A !"d >∈ 0

E-a%5#e 1.<

*e$ A be $&e pos#$#ve "u)bers !"d le$ 0 be $&e "e'!$#ve "u)bers. T&e" A !"d 0 !re d#s,o#"$ s#"%e "o "u)ber #s bo$& pos#$#ve !"d "e'!$#ve.

E-a%5#e 1.3<

*e$ E L N6 y  !"d  L Nr s $ T&e" E !"d  !re d#s,o#"$.

3.3

Venn=E!#e$ D"ag$a%s

A s#)ple !"d #"s$ru%$#ve 9!y of #llus$r!$#"' $&e rel!$#o"s&#ps be$9ee" se$s #s #" $&e use of $&e so/%!lled 5e"/Euler d#!'r!)s or s#)ply 5e"" . d#!'r!)s. Here 9e represe"$ ! se$ by ! s#)ple pl!"e !re! usu!lly bou"ded by ! %#r%le. E-a%5#e 13.1<

Suppose A ⊂ 0 !"d s!y A ≠ 0 $&e" A !"d 0 %!" be des%r#bed by e#$&er d#!'r!):

A 0 E-a%5#e 13.<

0 A

Suppose A !"d 0 !re "o$ %o)p!r!ble. T&e" A !"d 0 %!" be represe"$ed by $&e d#!'r!) o" $&e r#'&$ #f $&ey !re d#s,o#"$ or $&e d#!'r!) o" $&e le$ #f $&ey !re "o$ d#s,o#"$. 11

MTH 131


A

A

0

A E-a%5#e 13.3<

*e$ A L N! b % d !"d 0L N% d e f. T&e" 9e #llus$r!$e $&ese se$s ! 5e"" d#!'r!) of $&e for): A

3. 6

!

%

0 e

b

d

f

A-"o%at"( De)e#o5%ent O0 Set Teo$,

I" !" !6#o)!$#% develop)e"$ of ! br!"%& of )!$&e)!$#%s o"e be'#"s 9#$&: 1. 7. 3.

U"def#"ed $er)s U"def#"ed rel!$#o"s A6#o)s rel!$#"' $&e u"def#"ed $er)s !"d u"def#"ed rel!$#o"s.

T&e" o"e develops $&eore)s b!sed upo" $&e !6#o)s !"d def#"#$#o"s E-a%5#e 16<1

1. 7. 3.

po#"$s !"d l#"es !re u"def#"ed $er)s po#"$s o" ! l#"e or eu#v!le"$ l#"e %o"$!#" ! po#"$ #s !" u"def#"ed rel!$#o" T9o of $&e !6#o)s !re: A6#o) 1: A6#o) 7:

17

I" !" !6#o)!$#% develop)e"$ of Pl!"e Eu%l#de!" 'eo)e$ry

T9o d#ffere"$ po#"$s !re o" o"e !"d o"ly o"e l#"e T9o d#ffere"$ l#"es %!""o$ %o"$!#" )ore $&!" o"e po#"$ #" %o))o".

MTH 131


I" !" !6#o)!$#% develop)e"$ of se$ $&eory: 1. 7. 3.

ele)e"$J !"d se$ !re u"def#"ed $er)s ele)e"$ belo"'s $o ! se$ #s u"def#"ed rel!$#o" T9o of $&e !6#o)s !re

A-"o% o0 E-tens"on<

T9o se$s A !"d 0 !re eu!l #f !"d o"ly #f every ele)e"$ #" A belo"'s $o 0 !"d every ele)e"$ #" 0 belo"'s $o A.

A-"o% o0 S5e("0"(at"on<

*e$ P6B be !"y s$!$e)e"$ !"d le$ A be !"y se$. T&e" $&ere e6#s$s ! se$: 0LN! ! ∈ A P!B #s $rue

Here P6B #s ! se"$e"%e #" o"e v!r#!ble for 9&#%& P!B #s $rue or f!lse for !"y ! ∈ A. for e6!)ple P6B %ould be $&e se"$e"%e 67 L 2 or 6 #s ! )e)ber of $&e U"#$ed !$#o"s 6.

CONCLUSION

ou &!ve bee" #"$rodu%ed $o b!s#% %o"%ep$s of se$s se$ "o$!$#o" e.$.% $&!$ 9#ll be bu#l$ upo" #" o$&er u"#$s. If you &!ve "o$ )!s$ered $&e) by "o9 you 9#ll "o$#%e you &!ve $o %o)e b!%+ $o $&#s u"#$ fro) $#)e $o $#)e. 7.

SUMMARY

A su))!ry of $&e b!s#% %o"%ep$ of se$ $&eory #s !s follo9s:    



A set #s !"y 9ell/def#"ed l#s$ %olle%$#o" or %l!ss of ob,e%$s. G#ve" ! se$ A 9#$& elements 138> $&e tabular form of represe"$#"' $&#s se$ #s A L N1 3 8 > T&e set-builder form of $&e s!)e se$ #s A L N6X 6 L 7"  1< ≤"≤3 G#ve" $&e se$  L N724?@. $&e"  #s s!#d $o be infinite s#"%e $&e %ou"$#"' pro%ess of #$s ele)e"$s 9#ll "ever %o)e $o !" e"d o$&er9#se #$ #s finite T9o se$s of A !"d 0 !re s!#d $o be equal #f $&ey bo$& &!ve $&e s!)e ele)e"$s 9r#$$e" A L 0 13

MTH 131  

  




T&e null set  ∅ %o"$!#"s "o ele)e"$s !"d #s ! subse$ of every se$ T&e se$ A #s ! subse$ of !"o$&er se$ 0 9r#$$e" A ⊂ 0 #f every ele)e"$ of A #s !lso !" ele)e"$ of 0 #.e. for every 6∈A $&e" 6∈0 If 0 ⊂ A !"d 0 ≠A $&e" 0 #s ! proper subset of A T9o se$s A !"d 0 !re %o)p!r!ble #f A ⊂ 0 !"d 0⊂ A T&e power set 7S of !"y se$ S #s $&e f!)#ly of !ll $&e subse$s of S T9o se$s A !"d 0 !re s!#d $o be d#s,o#"$ #f $&ey do "o$ &!ve !"y ele)e"$ #" %o))o" #.e $&e#r #"$erse%$#o" #s ! "ull se$

8.

TUTOR=MAR?ED ASSIGNMENTS

1.

Re9r#$e $&e follo9#"' s$!$e)e"$ us#"' se$ "o$!$#o": 1. 6 does "o$ belo"' $o A. 7. R #s ! superse$ of S 3. d #s ! )e)ber of E 2.  #s "o$ ! subse$ of G 8. H does "o$ #"%luded D.

7.

(&#%& of $&ese se$s !re eu!l: Nsrs$W

3.

(&#%& se$s !re f#"#$eW 1. 7. 3. 2. 8.

Nr$s Ns$rs N$s$r

T&e )o"$&s of $&e ye!r N173@@== 1<< T&e people l#v#"' o" $&e e!r$& N6 X 6 #s eve" N173@@..B

T&e f#rs$ $&ree se$ !re f#"#$e. Al$&ou'& p&ys#%!lly #$ )#'&$ be #)poss#ble $o %ou"$ $&e "u)ber of people o" $&e e!r$& $&e se$ #s s$#ll f#"#$e. T&e l!s$ $9o se$ !re #"f#"#$e. If 9e ever $ry $o %ou"$ $&e eve" "u)bers 9e 9ould "ever %o)e $o $&e e"d. 2. 12

(&#%& 9ord #s d#ffere"$ fro) e!%& o$&er !"d 9&y: 1B e)p$y 7B vo#d 3Bero 2B "ullW

MTH 131


8

*e$ A L N6 y. Ho9 )!"y subse$s does A %o"$!#" !"d 9&!$ !re $&eyW

9.

REFERENCE AND FURTHER READING

Sey)our *#ps%&u$F S%&!u)Js Ou$l#"e Ser#es: T&eory !"d Proble)s of Se$ T&eory !"d rel!$ed $op#%s 1=42pp. 1  133. Su"d!y O. Iy!&e"F I"$rodu%$#o" $o Re!l A"!lys#s Re!l/v!lued fu"%$#o"s of ! re!l v!r#!ble 1==? 5ol. 1B

18

MTH 131

UNIT 


ASIC SET OPERATIONS

CONTENTS

1.< 7.<

2.< 8.< 4.< >.<

I"$rodu%$#o" Ob,e%$#ves M!#" 0ody 3.1 Se$ Oper!$#o"s 3.1.1 U"#o" 3.1.7 I"$erse%$#o" 3.1.3 D#ffere"%e 3.1.2 Co)ple)e"$ 3.7 Oper!$#o"s o" Co)p!r!ble Se$s Co"%lus#o" Su))!ry Tu$or M!r+ed Ass#'")e"$s Refere"%es !"d ur$&er Re!d#"'s

1.

INTRODUCTION

3.<

I" $&#s u"#$ 9e s&!ll see oper!$#o"s perfor)ed o" se$s !s #" s#)ple !r#$&)e$#%. T&#s oper!$#o"s s#)ply '#ve se$s ! l!"'u!'e of $&e#r o9". ou 9#ll "o$#%e #" subseue"$ u"#$s $&!$ you %!""o$ $!l+ of se$s 9#$&ou$ refere"%e sor$ of $o $&ese oper!$#o"s. .

O4ECTIVES

A$ $&e e"d of $&#s u"#$ you s&ould be !ble $o:  

Co)p!re $9o se$s !"dor !ss#'" $o $&e) !"o$&er se$ depe"d#"' o" $&e#r %o)p!r!b#l#$y. Represe"$ $&ese rel!$#o"s&#ps o" $&e 5e"" d#!'r!).

3.

MAIN ODY

3.1

Set O5e$at"ons

I" !r#$&)e$#% 9e le!r" $o !dd sub$r!%$ !"d )ul$#ply $&!$ #s 9e !ss#'" $o e!%& p!#r of "u)bers 6 !"d y ! "u)ber 6  y %!lled $&e su) of 6 !"d y ! "u)ber 6  y %!lled $&e d#ffere"%e of 6 !"d y !"d ! "u)ber 6y 14

MTH 131


%!lled $&e produ%$ of 6 !"d y. T&ese !ss#'")e"$s !re %!lled $&e oper!$#o"s of !dd#$#o" sub$r!%$#o" !"d )ul$#pl#%!$#o" of "u)bers. I" $&#s u"#$ 9e def#"e $&e oper!$#o" Union, Intersection !"d difference of se$s $&!$ #s 9e 9#ll !ss#'" "e9 p!#rs of se$s A !"d 0. I" ! l!$er u"#$ 9e 9#ll see $&!$ $&ese se$ oper!$#o"s be&!ve #" ! )!""er so)e9&!$ s#)#l!r $o $&e !bove oper!$#o"s o" "u)bers. 3.1.1 Un"on

T&e u"#o" of se$s A !"d 0 #s $&e se$ of !ll ele)e"$s 9&#%& belo"' $o A or $o 0 or $o bo$&. (e de"o$e $&e u"#o" of A !"d 0 byF A ∪ 0 (&#%& #s usu!lly re!d A u"#o" 0 E-a%5#e 1.1<

I" $&e 5e"" d#!'r!) #" f#' 7/1 9e &!ve s&!ded A ∪ 0 #.e. $&e !re! of A !"d $&e !re! of 0.

A

0

A ∪ 0 #s s&!ded #' 7.1 E-a%5#e 1.<

*e$ S L N! b. %. d !"d T L Nf b d '. T&e" S T L N! b % d f '.

E-a%5#e 1.3<

*e$ P be $&e se$ of pos#$#ve re!l "u)bers !"d le$ Y be $&e se$ of "e'!$#ve re!l 1>

MTH 131


"u)bers. T&e P ∪ Y $&e u"#o" of P !"d Y %o"s#s$ of !ll $&e re!l "u)bers e6%ep$ ero. T&e u"#o" of A !"d 0 )!y !lso be def#"ed %o"%#sely by A ∪ 0 L N6 6 ∈ A or 6 ∈ 0 Re%a$* .1< I$

follo9s d#re%$ly fro) $&e def#"#$#o" of $&e u"#o" of $9o se$s $&!$ A ∪ 0 !"d 0 ∪ A !re $&e s!)e se$ #.e. A ∪ 0 L 0 ∪ A

Re%a$* .< 0o$&

A !"d 0 !re !l9!ys subse$s of A !"d 0 $&!$

#s A ⊂ A ∪ 0B !"d 0 ⊂ A ∪ 0B

I" so)e boo+s $&e u"#o" of A !"d 0 #s de"o$ed by A  0 !"d #s %!lled $&e se$/$&eore$#% su) of A !"d 0 or s#)ply A plus 0. 3.1. Inte$se(t"on

T&e Intersection of se$s A !"d 0 #s $&e se$ of ele)e"$s 9&#%& !re %o))o" $o A !"d 0 $&!$ #s $&ose ele)e"$s 9&#%& belo"' $o A !"d 9&#%& belo"' $o 0. (e de"o$e $&e #"$erse%$#o" of A !"d 0 by: (&#%& #s re!d A #"$erse%$#o" 0. E-a%5#e .1<

1?

A ∩ 0

I" $&e 5e"" d#!'r!) #" f#' 7.7 9e &!ve s&!ded A ∩ 0 $&e !re! $&!$ #s %o))o" $o bo$& A !"d 0.

MTH 131


A ∩ 0 #s s&!ded #' 7.7 E-a%5#e .<

*e$ S L N! b % d !"d T L Nf b d '. T&e" S ∩ T L Nb d

E-a%5#e .3<

*e$ 5 L 7 3 4 @@ #.e. $&e )ul$#ples of 7F !"d le$ ( L N3 4 =@. #.e. $&e )ul$#ples of 3. T&e" 5 ∩ ( L N4 17 1?@@

T&e #"$erse%$#o" of A !"d 0 )!y !lso be def#"ed %o"%#sely by A ∩ 0 L N6  6 ∈ A 6 ∈ 0 Here $&e %o))! &!s $&e s!)e )e!"#"' !s !"d. Re%a$* .3< I$

of $9o se$s $&!$F

follo9s d#re%$ly fro) $&e def#"#$#o" of $&e #"$erse%$#o" A ∩ 0 L 0 ∩ A

Re%a$* .6< E!%& of $&e se$s A !"d 0 %o"$!#"s A

∩ 0 !s ! subse$ #.e.

A ∩ 0B ⊂ A !"d A ∩ 0B ⊂ 0

1=

MTH 131


Re%a$* .7< If se$s A !"d 0 &!ve "o ele)e"$s #" %o))o" #.e. #f

A !"d 0 !re d#s,o#"$ $&e" $&e #"$erse%$#o" of A !"d 0 #s $&e "ull se$ #.e. A ∩ 0 L ∅.

I" so)e boo+s espe%#!lly o" prob!b#l#$y $&e #"$erse%$#o" of A !"d 0 #s de"o$ed by A0 !"d #s %!lled $&e se$/$&eore$#% produ%$ of A !"d 0 or s#)ply A $#)es 0. 3.1.3 DIFFERENCE

T&e d#ffere"%e of se$s A !"d 0 #s $&e se$ of ele)e"$s 9&#%& belo"' $o A bu$ 9&#%& do "o$ belo"' $o 0. (e de"o$e $&e d#ffere"%e of A !"d 0 by A0 (&#%& #s re!d A d#ffere"%e 0 or s#)ply A )#"us 0. E-a%5#e 3.1<

I" $&e 5e"" d#!'r!) #" #' 7.3 9e &!ve s&!ded A  0 $&e !re! #" A 9&#%& #s "o$ p!r$ of 0.

A  0 #s s&!ded #' 7.3 E-a%5#e 3.<

*e$ R be $&e se$ of re!l "u)bers !"d le$ Y be $&e se$ of r!$#o"!l "u)bers. T&e" R  Y %o"s#s$s of $&e #rr!$#o"!l "u)bers.

T&e d#ffere"%e of A !"d 0 )!y !lso be def#"ed %o"%#sely by A  0 L N 6  6 ∈ A 6 ∉ 0 7<

MTH 131


Re%a$* .8< Se$ A %o"$!#"s A  0 !s ! subse$ #.e.

A  0B ⊂ A se$s A  0B A ∩ 0 !"d 0  AB !re )u$u!lly d#s,o#"$ $&!$ #s $&e #"$erse%$#o" of !"y $9o #s $&e "ull se$.

Re%a$* .9< T&e

T&e d#ffere"%e of A !"d 0 #s so)e$#)es de"o$ed by A0 or A Z 0 3.1.6 Co%5#e%ent

T&e %o)ple)e"$ of ! se$ A #s $&e se$ of ele)e"$s $&!$ do "o$ belo"' $o A $&!$ #s $&e d#ffere"%e of $&e u"#vers!l se$ U !"d A. (e de"o$e $&e %o)ple)e"$ of A by A′ E-a%5#e 6.1<

I" $&e 5e"" d#!'r!) #" #' 7.2 9e s&!ded $&e %o)ple)e"$ of A #.e. $&e !re! ou$s#de A. Here 9e !ssu)e $&!$ $&e u"#vers!l se$ U %o"s#s$s of $&e !re! #" $&e re%$!"'le.

A

0

A@ #s s&!ded

#'. 7.2

E-a%5#e 6.<

E-a%5#e 6.3<

*e$ $&e U"#vers!l se$ U be $&e E"'l#s& !lp&!be$ !"d le$ T L N! b %. T&e"F TJ L Nd e f @.. y  *e$ E L N7 2 4 @ $&!$ #s $&e eve" "u)bers. T&e" E′ L N1 3 8 @ $&e odd "u)bers. Here 9e 71

MTH 131


!ssu)e $&!$ $&e u"#vers!l se$ #s $&e "!$ur!l "u)bers 1 7 3@.. T&e %o)ple)e"$ of A )!y !lso be def#"ed %o"%#sely byF A′ L N6 6 ∈ U 6 ∉ A or s#)ply A′ L N6  6 ∉ A (e s$!$e so)e f!%$s !bou$ se$s 9&#%& follo9 d#re%$ly fro) $&e def#"#$#o" of $&e %o)ple)e"$ of ! se$. Re%a$* .:< T&e u"#o" of !"y se$ A

u"#vers!l se$ #.e.

!"d #$s %o)ple)e"$ A ′ #s $&e

A ∪ AJ L U ur$&er)ore se$ A !"d #$s %o)ple)e"$ A ′ !re d#s,o#"$ #.e. A ∩ AJ L ∅ Re%a$* .;<

T&e %o)ple)e"$ of $&e u"#vers!l se$ U #s $&e "ull se$ ∅ !"d v#%e vers! $&!$ #s UJ L ∅ !"d ∅J L U

Re%a$* .1< T&e %o)ple)e"$ of $&e %o)ple)e"$ of se$ A

#$self. More br#efly

#s $&e se$ A

A′B′ L A Our "e6$ re)!r+ s&o9s &o9 $&e d#ffere"%e of $9o se$s %!" be def#"ed #" $er)s of $&e %o)ple)e"$ of ! se$ !"d $&e #"$erse%$#o" of $9o se$s. More spe%#f#%!lly 9e &!ve $&e follo9#"' b!s#% rel!$#o"s&#p: Re%a$* .11
A !"d $&e %o)ple)e"$ of 0 $&!$ #s

#"$erse%$#o" of

A  0 L A ∩ 0′ T&e proof of Re)!r+ 7.11 follo9s d#re%$ly fro) def#"#$#o"s: 77

MTH 131


A  0 L N6  6∈A 6∉0 L N6  6∈A 6∉0J L A ∩ 0J 3.

O5e$at"ons on Co%5a$a&#e Sets

T&e oper!$#o"s of u"#o" #"$erse%$#o" d#ffere"%e !"d %o)ple)e"$ &!ve s#)ple proper$#es 9&e" $&e se$s u"der #"ves$#'!$#o" !re %o)p!r!ble. T&e follo9#"' $&eore)s %!" be proved. *e$ A be ! subse$ of 0. T&e" $&e u"#o" #"$erse%$#o" of A !"d 0 #s pre%#sely A $&!$ #s

Teo$e% .1<

A ⊂ 0 #)pl#es A ∩ 0 L A *e$ A be ! subse$ of 0. T&e" $&e of A !"d 0 #s pre%#sely 0 $&!$ #s

Teo$e% .<

A ⊂ 0 #)pl#es A ∪ 0 L 0 *e$ A be ! subse$ of 0. T&e" 0J #s ! subse$ of AJ $&!$ #s

Teo$e% .3<

A ⊂ 0 #)pl#es 0J ⊂ AJ (e #llus$r!$e T&eore) 7.3 by $&e 5e"" d#!'r!)s #" #' 7/8 !"d 7/4. o$#%e &o9 $&e !re! of 0J #s #"%luded #" $&e !re! of AJ. 0 0

A

A

0′ #s s&!ded #' 7.8

A′ #s s&!ded #' 7.4

73

MTH 131


Teo$e% .6<

*e$ A be ! subse$ of 0. T&e" $&e U"#o" of A !"d 0  AB #s pre%#sely 0 $&!$ #s A ⊂ 0 #)pl#es A ∪ 0  AB L 0

E-e$("ses

1.

I" $&e 5e"" d#!'r!) belo9 s&!de A U"#o" 0 $&!$ #s A ∪ 0:

!B

bB

%B

So#!t"on<

T&e u"#o" of A !"d 0 #s $&e se$ of !ll ele)e"$s $&!$ belo"' $o A !"d $o 0 or $o bo$&. (e $&erefore s&!de $&e !re! #" A !"d 0 !s follo9s:

7.

*e$ A L N1732 0 L N724? !"d C L N3284. #"d !B A ∪0 bB A ∪ C %B 0 ∪ C dB 0 ∪ 0.

So#!t"on<

To for) $&e u"#o" of A !"d 0 9e pu$ !ll $&e ele)e"$s fro) A $o'e$&er 9#$& $&e ele)e"$s of 0 A%%ord#"'ly A ∪ 0 L N17324? 72

MTH 131


A ∪ C L N 173284 0 ∪ C L N724?38 0 ∪ 0 L N724? o$#%e $&!$ 0∪0 #s pre%#sely 0. 3.

*e$ A 0 !"d C be $&e se$s #" Proble) 7. #"d 1B  A ∪ 0B ∪ C 7B A ∪ 0 ∪ CB.

So#!t"on<

1.

(e f#rs$ f#"d A ∪ 0B L N17324?. T&e" $&e u"#o" of NA ∪ 0 !"d C #s A∪0B ∪C L N17324?8

7.

(e f#rs$ f#"d 0∪CB L N724?38. T&e" $&e u"#o" of A !"d 0∪CB #s A ∪ 0 ∪ CB L N17324?8. o$#%e $&!$ A ∪0B ∪ C L A ∪0 ∪ CB.

6.

CONCLUSION

ou &!ve see" &o9 $&e b!s#% oper!$#o"s of U"#o" I"$erse%$#o" D#ffere"%e !"d Co)ple)e"$ o" se$s 9or+ l#+e $&e oper!$#o"s o" "u)bers. T&ese !re !lso $&e b!s#% sy)bols !sso%#!$ed 9#$& se$ $&eory. 7.

SUMMARY

T&e b!s#% se$ oper!$#o"s !re U"#o" I"$erse%$#o" D#ffere"%e !"d Co)ple)e"$ def#"ed !s:  



T&e Union of se$s A !"d 0 de"o$ed by A ∪  #s $&e se$ of !ll ele)e"$s 9&#%& belo"' $o A or $o 0 or $o bo$&. T&e intersection of se$s A !"d 0 de"o$ed by A ∩  #s $&e se$ of ele)e"$s 9&#%& !re %o))o" $o A !"d 0. If A !"d 0 !re d#s,o#"$ $&e" $&e#r #"$erse%$#o" #s $&e ull se$ ∅. T&e difference of se$s A !"d 0 de"o$ed by A  B #s $&e se$ of ele)e"$s 9&#%& belo"' $o A bu$ 9&#%& do "o$ belo"' $o 0. 78

MTH 131 

8.

T&e complement of ! se$ A de"o$ed by A@ #s $&e se$ of ele)e"$s 9&#%& do "o$ belo"' $o A $&!$ #s $&e d#ffere"%e of $&e u"#vers!l se$ U !"d A.

TUTOR  MAR?ED ASSIGNMENTS

1. 7. 3. 2.

8. 9.


*e$  L NTo) D#%+ H!rry  L NTo) M!r% Er#% !"d [ L NM!r% Er#% Ed9!rd. #"d !B  ∪  bB  ∪ [ %B  ∪ [ Prove: A ∩ ∅ L ∅. Prove Re)!r+ 7.4: A  0B ⊂ A. *e$ U L N173@?= A L N1732 0 L N724? !"d C L N3284. #"d !B AJ bB 0J %B A ∩ CBJ dB A ∪ 0BJ eB AJBJ fB 0  CBJ Prove: 0  A #s ! subse$ of AJ.

REFERENCES AND FURTHER READINGS

Sey)our *#ps%&u$F S%&!u)Js Ou$l#"e Ser#es: T&eory !"d Proble)s of Se$ T&eory !"d rel!$ed $op#%s 1=42 pp. 1  133. Su"d!y O. Iy!&e"F I"$rodu%$#o" $o Re!l A"!lys#s Re!l  v!lued fu"%$#o"s of ! re!l v!r#!bleB 1==? 5ol. 1

74

MTH 131

UNIT 3


SET OF NUMERS

CONTENTS

1.< 7.< 3.<

2.< 8.< 4.< >.<

I"$rodu%$#o" Ob,e%$#ves M!#" 0ody 3.1 Se$ Oper!$#o"s 3.1.1 I"$e'ers [ 3.1.7 R!$#o"!l "u)bers  3.1.3 !$ur!l u)bers N 3.1.2 Irr!$#o"!l u)bers @ 3.1.8 *#"e d#!'r!) of $&e u)ber sys$e)s 3.7 De%#)!ls !"d Re!l u)bers 3.3 I"eu!l#$#es 3.2 Absolu$e 5!lue 3.8 I"$erv!ls 3.8.1 Proper$#es of #"$erv!ls 3.8.7 I"f#"#$e I"$erv!ls 3.4 0ou"ded !"d U"bou"ded Se$s Co"%lus#o" Su))!ry Tu$or  M!r+ed Ass#'")e"$s Refere"%es !"d ur$&er Re!d#"'s

1.

INTRODUCTION

Al$&ou'& $&e $&eory of se$s #s very 'e"er!l #)por$!"$ se$s 9&#%& 9e )ee$ #" ele)e"$!ry )!$&e)!$#%s !re se$s of "u)bers. Of p!r$#%ul!r #)por$!"%e espe%#!lly #" !"!lys#s #s $&e se$ of real numbers, 9&#%& 9e de"o$e by

I" f!%$ 9e !ssu)e #" $&#s u"#$ u"less o$&er9#se s$!$ed $&!$ $&e se$ of re!l "u)bers #s ou$ u"#vers!l se$. (e f#rs$ rev#e9 so)e ele)e"$!ry proper$#es of re!l "u)bers before !pply#"' our e le)e"$!ry pr#"%#ples of se$ $&eory $o se$s of "u)bers. T&e se$ of re!l "u)bers !"d #$s proper$#es #s %!lled $&e real number system. 7>

MTH 131

.


O4ECTIVES

Af$er s$udy#"' $&#s u"#$ you s&ould be !ble $o do $&e follo9#"':  Represe"$ $&e se$ of "u)bers o" $&e re!l l#"e  Perfor) $&e b!s#% se$ oper!$#o"s o" #"$erv!ls 3.

MAIN ODY

3.1

Rea# N!%&e$sB

O"e of $&e )os$ #)por$!"$ proper$#es of $&e re!l "u)bers #s $&!$ po#"$s o" ! s$r!#'&$ l#"e $&!$ %!" represe"$ $&e). As #" #' 3.1 9e %&oose ! po#"$ %!lled $&e or#'#" $o represe"$ < !"d !"o$&er po#"$ usu!lly $o $&e r#'&$ $o represe"$ 1. T&e" $&ere #s ! "!$ur!l 9!y $o p!#r off $&e po#"$s o" $&e l#"e !"d $&e re!l "u)bers $&!$ #s e!%& po#"$ 9#ll represe"$ ! u"#ue re!l "u)ber !"d e!%& re!l "u)ber 9#ll be represe"$ed by ! u"#ue po#"$. (e refer $o $&#s l#"e !s $&e real line. A%%ord#"'ly 9e %!" use $&e 9ords po#"$ !"d "u)ber #"$er%&!"'e!bly. T&ose "u)bers $o $&e r#'&$ of < #.e. o" $&e s!)e s#de !s 1 !re %!lled $&e positive numbers !"d $&ose "u)bers $o $&e lef$ of < !re %!lled $&e negative numbers. T&e "u)ber < #$self #s "e#$&er pos#$#ve "or "e'!$#ve. e L 7. >1? \

/

/8

/2

/3

/7

/1

<

1

7

3

#' 3.1 3.1. Intege$sB 

T&e #"$e'ers !re $&ose re!l "u)bers @ /3 /7 /1 < 1 7 3@ (e de"o$e $&e #"$e'ers by [F &e"%e 9e %!" 9r#$e [ L N @ /7 / 1 < 1 7@ T&e #"$e'ers !re !lso referred $o !s $&e 9&ole "u)bers. 7?

2

8

MTH 131


O"e #)por$!"$ proper$y of $&e #"$e'ers #s $&!$ $&ey !re %losed u"der $&e oper!$#o"s of !dd#$#o" )ul$#pl#%!$#o" !"d sub$r!%$#o"F $&!$ #s $&e su) produ%$ !"d d#ffere"%e of $9o #"$e'ers #s !'!#" #" #"$e'er. o$#%e $&!$ $&e uo$#e"$ of $9o #"$e'ers e.'. 3 !"d > "eed "o$ be !" #"$e'erF &e"%e $&e #"$e'ers !re "o$ %losed u"der $&e oper!$#o" of d#v#s#o". 3.1.3 Rat"ona# N!%&e$sB 

T&e rational numbers !re $&ose re!l "u)bers 9&#%& %!" be e6pressed !s $&e r!$#o of $9o #"$e'ers. (e de"o$e $&e se$ of r!$#o"!l "u)bers by Y. A%%ord#"'ly Y L N6  6 L p  9&ere p ∈   ∈  o$#%e $&!$ e!%& #"$e'er #s !lso ! r!$#o"!l "u)ber s#"%e for e6!)ple 8 L 81F &e"%e [ #s ! subse$ of Y. T&e r!$#o"!l "u)bers !re %losed "o$ o"ly u"der $&e oper!$#o"s of !dd#$#o" )ul$#pl#%!$#o" !"d sub$r!%$#o" bu$ !lso u"der $&e oper!$#o" of d#v#s#o" e6%ep$ by
T&e natural numbers !re $&e pos#$#ve #"$e'ers. (e de"o$e $&e se$ of "!$ur!l "u)bers by F &e"%e  L N173@.. T&e "!$ur!l "u)bers 9ere $&e f#rs$ "u)ber sys$e) developed !"d 9ere used pr#)!r#ly !$ o"e $#)e for %ou"$#"'. o$#%e $&e follo9#"' rel!$#o"s&#p be$9ee" $&e !bove "u)bers sys$e)s:  ⊂ [ ⊂ Y ⊂ T&e "!$ur!l "u)bers !re %losed o"ly u"der $&e oper!$#o" of !dd#$#o" !"d )ul$#pl#%!$#o". T&e d#ffere"%e !"d uo$#e"$ of $9o "!$ur!l "u)bers "eeded "o$ be ! "!$ur!l "u)ber. T&e prime numbers !re $&ose "!$ur!l "u)bers p e6%lud#"' 1 9&#%& !re o"ly d#v#s#ble 1 !"d p #$self. (e l#s$ $&e f#rs$ fe9 pr#)e "u)bers: 738>11131>1=@ 7=

MTH 131


3.1.7 I$$at"ona# N!%&e$sB @

T&e #rr!$#o"!l "u)bers !re $&ose re!l "u)bers 9&#%& !re "o$ r!$#o"!l $&!$ #s $&e se$ of #rr!$#o"!l "u)bers #s $&e %o)ple)e"$ of $&e se$ of r!$#o"!l "u)bers Y #" $&e re!l "u)bers F &e"%e YJ de"o$e $&e #rr!$#o"!l "u)bers. E6!)ples of #rr!$#o"!l "u)bers !re √3 π √7 e$%. 3.1.8 L"ne D"ag$a% o0 te N!%&e$ S,ste%s

#' 3 /7 belo9 #s ! l#"e d#!'r!) of $&e v!r#ous se$s of "u)ber 9&#%& 9e &!ve #"ves$#'!$ed. or %o)ple$e"ess $&e d#!'r!) #"%lude $&e se$s of %o)ple6 "u)bers "u)ber of $&e for) !  b# 9&ere ! !"d b !re re!l. o$#%e $&!$ $&e se$ of %o)ple6 "u)bers #s superse$ of $&e se$ of re!l "u)bers.B Co)ple6 u)bers Re!l u)bers R!$#o"!l u)bers

Irr!$#o"!l u)bers

I"$e'ers e'!$#ve I"$e'ers

[ero

!$ur!l u)bers Pr#)e u)bers

#' 3.7 3.

De("%a#s an+ Rea# N!%&e$s

Every re!l "u)ber %!" be represe"$ed by ! "o"/$er)#"!$#"' de%#)!l. T&e de%#)!l represe"$!$#o" of ! r!$#o"!l "u)ber p %!" be fou"d by 3<

MTH 131


d#v#d#"' $&e de"o)#"!$or  #"$o $&e "u)er!$or p. If $&e #"d#%!$ed d#v#s#o" $er)#"!$es !s for (e 9r#$e Or

3? L .3>8 3? L .3>8<<< 3? L .3>2===@

If 9e #"d#%!$ed d#v#s#o" of  #"$o p does "o$ $er)#"!$e $&e" #$ #s +"o9" $&!$ ! blo%+ of d#'#$s 9#ll %o"$#"u!lly be repe!$edF for e6!)ple 711 L .1?1?1?@ (e "o9 s$!$e $&e b!s#% f!%$ %o""e%$#"' de%#)!ls !"d re!l "u)bers. T&e r!$#o"!l "u)bers %orrespo"d pre%#sely $o $&ose de%#)!ls #" 9&#%& ! blo%+ of d#'#$s #s %o"$#"u!lly repe!$ed !"d $&e #rr!$#o"!l "u)bers %orrespo"d $o $&e o$&er "o"/$er)#"!$#"' de%#)!ls. 3.3

Ine>!a#"t"es

T&e %o"%ep$ of order #s #"$rodu%ed #" $&e re!l "u)ber sys$e) by $&e De0"n"t"on<

T&e re!l "u)ber ! #s less $&!" $&e re!l "u)ber b 9r#$$e" ! ] b

If b  ! #s ! pos#$#ve "u)ber. T&e follo9#"' proper$#es of $&e rel!$#o" ! ] b %!" be prove". *e$ ! b !"d % be re!l "u)bersF $&e": P1: P7: P3: P2: P8:

E#$&er ! ] b ! L b or b ] !. If ! ] b !"d b ] % $&e" ! ] %. If ! ] b $&e" !  % ] b  % If ! ] b !"d % #s pos#$#ve $&e" !% ] b% If ! ] b !"d % #s "e'!$#ve $&e" b% ] !%.

Geo)e$r#%!lly #f ! ] b $&e" $&e po#"$ ! o" $&e re!l l#"e l#es $o $&e lef$ of $&e po#"$ b. (e !lso de"o$e ! ] b by

b^!

31

MTH 131


(&#%& re!ds b #s greater then !. ur$&er)ore 9e 9r#$e ! ] b or b ^ ! #f ! ] b or ! L b $&!$ #s #f ! #s "o$ 're!$er $&!" b. E-a%5#e 1.1<

7 ] 8F /4 ] /3 !"d 2 ] 2F 8 ^ /?

E-a%5#e 1.<

T&e "o$!$#o" 6 ] 8 )e!"s $&!$ 6 #s ! re!l "u)ber 9&#%& #s less $&!" 8F &e"%e 6 l#es $o $&e lef$ of 8 o" $&e re!l l#"e T&e "o$!$#o" 7 ] 6 ] >F )e!"s 7 ] 6 !"d !lso 6 ] >F &e"%e 6 9#ll l#e be$9ee" 7 !"d > o" $&e re!l l#"e.

Re%a$* 3.1< o$#%e

$&!$ $&e %o"%ep$ of order #.e. $&e rel!$#o" ! ] b #s def#"ed #" $er)s of $&e %o"%ep$ of pos#$#ve "u)bers. T&e fu"d!)e"$!l proper$y of $&e pos#$#ve "u)bers 9&#%& #s used $o prove proper$#es of $&e rel!$#o" ! ] b #s $&!$ $&e pos#$#ve "u)bers !re %losed u"der $&e oper!$#o"s of !dd#$#o" !"d )ul$#pl#%!$#o". Moreover $&#s f!%$ #s #"$#)!$ely %o""e%$ed 9#$& $&e f!%$ $&!$ $&e "!$ur!l "u)bers !re !lso %losed u"der $&e oper!$#o"s of !dd#$#o" !"d )ul$#pl#%!$#o".

Re%a$* 3.< T&e

follo9#"' s$!$e)e"$s !re $rue 9&e" ! b % !re !"y re!l "u)bers: 1. 7. 3.

3.6

!]! #f ! ] b !"d b ] ! $&e" ! L b. #f ! ] b !"d b ] % $&e" ! ] %.

A&so#!te Va#!e

T&e !bsolu$e v!lue of ! re!l "u)ber 6 de"o$ed by  6 #s def#"ed by $&e for)ul!  6 37

L

6 #f 6 ^ < /6 #f 6 ] <

MTH 131


$&!$ #s #f 6 #s pos#$#ve or ero $&e"  6 eu!ls 6 !"d #f 6 #s "e'!$#ve $&e"  6 eu!ls  6. Co"seue"$ly $&e !bsolu$e v!lue of !"y "u)ber #s !l9!ys "o"/"e'!$#ve #.e.  6 ^ < for every 6 ∈ ℜ. Geo)e$r#%!lly spe!+#"' $&e !bsolu$e v!lue of 6 #s $&e d#s$!"%e be$9ee" $&e po#"$ 6 o" $&e re!l l#"e !"d $&e or#'#" #.e. $&e po#"$ <. Moreover $&e d#s$!"%e be$9ee" !"y $9o po#"$s #.e. re!l "u)bers ! !"d b #s  ! / b L  b / ! . E-a%5#e .1<

 /7 L 7  > L >.  /π L π

E-a%5#e .<

T&e s$!$e)e"$  6 ] 8 %!" be #"$erpre$ed $o )e!" $&!$ $&e d#s$!"%e be$9ee" 6 !"d $&e or#'#" #s less $&!" 8 #.e. 6 )us$ l#es be$9ee" /8 !"d 8 o" $&e re!l l#"e. I" o$&er 9ords  6 ] 8 !"d /8 ] 6 ] 8

&!ve #de"$#%!l )e!"#"'. S#)#l!rly  6 ] 8 !"d /8 ] 6 ] 8

&!ve #de"$#%!l )e!"#"'. 3.7

Inte$)a#s

Co"s#der $&e follo9#"' se$ of "u)bersF A1 L N6 7 ] 6 ] 8 A7 L N6 7 ] 6 ] 8 A3 L N6 7 ] 6 ] 8 A2 L N6 7 ] 6 ] 8 o$#%e $&!$ $&e four se$s %o"$!#" o"ly $&e po#"$s $&!$ l#e be$9ee" 7 !"d 8 9#$& $&e poss#ble e6%ep$#o"s of 7 !"dor 8. (e %!ll $&ese se$s #"$erv!ls $&e "u)bers 7 !"d 8 be#"' $&e e"dpo#"$s of e!%& #"$erv!l. Moreover A1 #s !" open interval !s #$ does "o$ %o"$!#" e#$&er e"d po#"$: 33

MTH 131


A7 #s ! closed interval !s #$ %o"$!#"s bo$&er e"dpo#"$sF A 3 !"d A2 !re open-closed !"d closed-open respe%$#vely. (e d#spl!y #.e. 'r!p& $&ese se$s o" $&e re!l l#"e !s follo9s.

/

/

/

/

/

/

/

/

/

<

1

A1

/

<

7

3

1

7

2

3

8

2

4

8

4

8

4

A7 /

/

/

/

/

<

1

7

3

2

A3

/

/

/

/

/

<

1

7

3

2

8

4

A2 o$#%e $&!$ #" e!%& d#!'r!) 9e %#r%le $&e e"dpo#"$s 7 !"d 8 !"d $&#%+e" or s&!deB $&e l#"e se')e"$ be$9ee" $&e po#"$s. If !" #"$erv!l #"%ludes !" e"dpo#"$ $&e" $&#s #s de"o$ed by s&!d#"' $&e %#r%le !bou$ $&e e"dpo#"$. S#"%e #"$erv!ls !ppe!r very of$e" #" )!$&e)!$#%s ! s&or$er "o$!$#o" #s freue"$ly used $o des#'"!$ed #"$erv!ls Spe%#f#%!lly $&e !bove #"$erv!ls !re so)e$#)es de"o$ed byF A1 L 7 8B A7 L _7 8` A3 L 7 8B A2 L _7 8B 32

MTH 131


o$#%e $&!$ ! p!re"$&es#s #s used $o des#'"!$e !" ope" e"dpo#"$ #.e. !" e"dpo#"$ $&!$ #s "o$ #" $&e #"$erv!l !"d ! br!%+e$ #s used $o des#'"!$e ! %losed e"dpo#"$. 3.7.1 P$o5e$t"es o0 Inte$)a#s

*e$ be $&e f!)#ly of !ll #"$erv!ls o" $&e re!l l#"e. (e #"%lude #" $&e "ull se$ ∅ !"d s#"'le po#"$s ! L _! !`. T&e" $&e #"$erv!ls &!ve $&e follo9#"' proper$#es: 1.

T&e #"$erse%$#o" of $9o #"$erv!ls #s !" #"$erv!l $&!$ #s A ∈ ℑ 0 ∈ ℑ #)pl#es A ∩ 0 ∈ ℑ

7.

T&e u"#o" of $9o "o"/d#s,o#"$ #"$erv!ls #s !" #"$erv!l $&!$ #s A ∈ ℑ 0 ∈ ℑ A ∩ 0 ≠ ∅ #)pl#es A ∪ 0 ∈ ℑ

3.

T&e d#ffere"%e of $9o "o"/%o)p!r!ble #"$erv!ls #s !" #"$erv!l $&!$ #s A ∈ ℑ 0 ∈ ℑ A ⊄ 0 0 ⊄ A #)pl#es A / 0 ∈ ℑ

E-a%5#e 3.1<

*e$ A L N7 2B 0 L 3 ?B. T&e" A ∩ 0 L 3 2B A ∪ 0 L _7 ?B A  0 L _7 3` 0  A L _2 ?B

3.7. In0"n"te Inte$)a#s

Se$s of $&e for) A L N6 6 ^ 1 0 L N6 6 ^ 7 C L N6 6 ] 3 D L N6 6 ] 2 E L N6  6 ∈ ℜ Are %!lled #"f#"#$e #"$erv!ls !"d !re !lso de"o$ed by A L 1 ∞ B 0 L _7 ∞ B C L  / ∞ 3B D L / ∞ 2` E L /∞ ∞B 38

MTH 131


(e plo$ $&ese #"$erv!ls o" $&e re!l l#"e !s follo9s: /2

/3

/7

/1

<

1

7

3

2

7

3

2

A #s s&!ded

/2

/3

/7

/1

<

1 0 #s s&!ded

/2

/3

/7

/1

< 1 C #s s&!ded

7

3

2

/2

/3

/7

/1

<

7

3

2

7

3

2

1 D #s s&!ded

/2 3.8

/3

/7

/1

<

1 E #s s&!ded

o!n+e+ an+ Un&o!n+e+ Sets

*e$ A be ! se$ of "u)bers $&e" A #s %!lled bounded se$ #f A #s $&e subse$ of ! f#"#$e #"$erv!l. A" eu#v!le"$ def#"#$#o" of bou"ded"ess #s De0"n"t"on 3.1<

Se$ A #s bounded #f $&ere e6#s$s ! pos#$#ve "u)ber M su%& $&!$  6  ≤ M.

for !ll 6 ∈ A. A se$ #s %!lled unbounded #f #$ #s "o$ bou"ded 34

MTH 131


o$#%e $&e" $&!$ A #s ! subse$ of $&e f#"#$e #"$erv!l _ / M M`. E-a%5#e 6.1<

*e$ A L N1 \  13@... T&e" A #s bou"ded s#"%e A #s %er$!#"ly ! subse$ of $&e %losed #"$erv!l _< 1`.

E-a%5#e 6.<

*e$ A L N7 2 4@... T&e" A #s !" u"bou"ded se$.

E-a%5#e 6.3<

*e$ A L N> 38< /2>3 7377 27. T&e" A #s bou"ded

Re%a$* 3.3<

If ! se$ A #s f#"#$e $&e" #$ #s "e%ess!r#ly bou"ded. If ! se$ #s #"f#"#$e $&e" #$ %!" be e#$&er bou"ded !s #" e6!)ple 2.1 or u"bou"ded !s #" e6!)ple 2.7

6.

CONCLUSION

T&e se$ of re!l "u)bers #s of u$)os$ #)por$!"%e #" !"!lys#s. All e6%ep$ $&e se$ of %o)ple6 "u)bersB o$&er se$s of "u)bers !re subse$s of $&e se$ of re!l "u)bers !s %!" be see" fro) $&e l#"e d#!'r!) of $&e "u)ber sys$e). 7.

SUMMARY

I" $&#s u"#$ you &!ve bee" #"$rodu%ed $o $&e se$s of "u)bers. T&e se$ of re!l "u)bers ℜ %o"$!#"s $&e se$ of #"$e'ers [ R!$#o"!l "u)bers Y !$ur!l "u)bers  !"d Irr!$#o"!l "u)bers YJ. I"$erv!ls o" $&e re!l l#"e !re ope" %losed ope"/%losed or %losed/ope" depe"d#"' o" $&e "!$ure of $&e e"dpo#"$s. 8.


1. 7.

Prove: If ! ] b !"d 0 ] % $&e" ! ] % U"der 9&!$ %o"d#$#o"s 9#ll $&e u"#o" of $9o d#s,o#"$ #"$erv!l be !" #"$erv!lW If $9o se$s R !"d S !re bou"ded 9&!$ %!" be s!#d !bou$ $&e u"#o" !"d #"$erse%$#o" of $&ese se$sW

3. 9.


3>

MTH 131



3?

MTH 131

UNIT 6


FUNCTIONS I

CONTENTS

1.< 7.< 3.<

2.< 8.< 4.< >.<

I"$rodu%$#o" Ob,e%$#ves M!#" body 3.1 Def#"#$#o" 3.7 M!pp#"'s Oper!$ors Tr!"sfor)!$#o"s 3.3 Eu!l fu"%$#o"s 3.2 R!"'e of ! fu"%$#o" 3.8 O"e  O"e fu"%$#o"s 3.4 O"$o fu"%$#o"s 3.> Ide"$#$y fu"%$#o" 3.? Co"s$!"$ u"%$#o"s 3.= Produ%$ u"%$#o" 3.=.1 Asso%#!$#v#$y of Produ%$s of u"%$#o"s 3.1< I"verse of ! fu"%$#o" 3.11 I"verse u"%$#o" 3.11.1 T&eore)s o" $&e I"verse u"%$#o" Co"%lus#o" Su))!ry Tu$or/M!r+ed Ass#'")e"$s Refere"%es !"d ur$&er Re!d#"'s

1.

INTRODUCTION

I" $&#s u"#$ you 9#ll be #"$rodu%ed $o $&e %o"%ep$ of fu"%$#o"s )!pp#"'s !"d $r!"sfor)!$#o"s. ou 9#ll !lso be '#ve" #"s$ru%$#ve !"d $yp#%!l e6!)ples of fu"%$#o"s. .

O4ECTIVES

A$ $&e e"d of $&#s u"#$ you s&ould be !ble $o:   

Ide"$#fy fu"%$#o"s fro) s$!$e)e"$s or d#!'r!)s S$!$e 9&e$&er ! fu"%$#o" #s o"e/o"e or o"$o #"d %o)pos#$#o" fu"%$#o" of $9o or )ore fu"%$#o"s

3=

MTH 131


3.

MAIN ODY

3.1

De0"n"t"on

Suppose $&!$ $o e!%& ele)e"$ #" ! se$ A $&ere #s !ss#'"ed by so)e )!""er or o$&er ! u"#ue ele)e"$ of ! se$ ℜ. (e %!ll su%& !ss#'")e"$ of function. If 9e le$ ƒ de"o$e $&ese !ss#'")e"$s 9e 9r#$eF f: A

0

9&#%& re!ds f #s ! fu"%$#o" of A o"$o 0. T&e se$ A #s %!lled $&e domain of $&e fu"%$#o" f !"d 0 #s %!lled $&e co-domain of f. ur$&er #f ! ∈ A $&e ele)e"$ #" 0 9&#%& #s !ss#'"ed $o ! #s %!lled $&e image of ! !"d #s de"o$ed byF ƒ!B

9&#%& re!ds f of !. (e l#s$ ! "u)ber of #"s$ru%$#ve e6!)ples of fu"%$#o"s. E-a%5#e 1.1<

*e$ f !ss#'" $o e!%& re!l "u)ber #$s su!re $&!$ #s for every re!l "u)ber 6 le$ f6B L 6 7. T&e do)!#" !"d %o/ do)!#" of f !re bo$& $&e re!l "u)bers so 9e %!" 9r#$e f:ℜ

ℜ

T&e #)!'e of /3 #s =F &e"%e 9e %!" !lso 9r#$e f/3B L = or f : 3  = E-a%5#e 1.<

*e$ f !ss#'" $o e!%& %ou"$ry #" $&e 9orld #$s %!p#$!l %#$y. Here $&e do)!#" of f #s $&e se$ of %ou"$r#es #" $&e 9orldF T&e %o/do)!#" of f #s $&e l#s$ of %!p#$!l %#$#es #" $&e 9orld. T&e #)!'e of r!"%e #s P!r#s $&!$ #s fr!"%eB L P!r#s

E-a%5#e 1.3<

*e$ A L N! b % d !"d 0 L N! b %. Def#"e ! fu"%$#o" f of A #"$o 0 by $&e %orrespo"de"%e f!B L b fbB L % f%B L % !"d fdB L b. 0y $&#s def#"#$#o" $&e #)!'e for e6!)ple of b #s %.

2<

MTH 131 E-a%5#e 1.6<


*e$ A L N/1 1. *e$ f !ss#'" $o e!%& r!$#o"!l "u)ber #" ℜ $&e "u)ber 1 !"d $o e!%& #rr!$#o"!l "u)ber #" ℜ $&e "u)ber /1. T&e" f: ℜ→A !"d f %!" be def#"ed %o"%#sely by f6B L 1 #f 6 #s r!$#o"!l /1 #f ! #s #rr!$#o"!l

E-a%5#e 1.7<

*e$ A L N! b % d !"d 0 L N6 y . *e$ f: A → 0 be def#"ed by $&e d#!'r!):

! b %

6 y 

d

o$#%e $&!$ $&e fu"%$#o"s #" e6!)ples 1.1 !"d 1.2 !re def#"ed by spe%#f#% for)ul!s. 0u$ $&#s "eed "o$ !l9!ys be $&e %!se !s #s #"d#%!$ed by $&e o$&er e6!)ples. T&e rules of %orrespo"de"%e 9&#%& def#"e fu"%$#o"s %!" be d#!'r!)s !s #" e6!)ple 1.8 %!" be 'eo'r!p&#%!l !s #" e6!)ple 1.7 or 9&e" $&e do)!#" #s f#"#$e $&e %orrespo"de"%e %!" be l#s$ed for e!%& ele)e"$ #" $&e do)!#" !s #" e6!)ple 1.2. 3.

Ma55"ngsB O5e$ato$sB T$ans0o$%at"ons

If A !"d 0 !re se$s #" 'e"er!l "o$ "e%ess!r#ly se$s of "u)bers $&e" ! fu"%$#o" f of A #"$o 0 #s freue"$ly %!lled ! )!pp#"' of A #"$o 0F !"d $&e "o$!$#o" f: : A ///^ 0 #s $&e" re!d f )!ps A #"$o 0. (e %!" !lso de"o$e ! )!pp#"' or fu"%$#o" f of A #"$o 0 by 21

MTH 131


ƒ

A

Or by $&e d#!'r!)

0

0

A

If $&e do)!#" !"d %o/do)!#" of ! fu"%$#o" !re bo$& $&e s!)e se$ s!y f: A ///^ A $&e" f #s freue"$ly %!lled !" operator or transformation o" A. As 9e 9#ll see l!$er oper!$ors !re #)por$!"$ spe%#!l %!ses of fu"%$#o"s. 3.3

E>!a# F!n(t"ons

If f !"d ' !re fu"%$#o"s def#"ed o" $&e s!)e do)!#" D !"d #f f!B L '!B for every ! ∈ D $&e" $&e fu"%$#o"s f !"d ' !re eu!l !"d 9e 9r#$e f L' E-a%5#e .1<

*e$ f6B L 67 9&ere 6 #s ! re!l "u)ber. *e$ '6B L 67 9&ere 6 #s ! %o)ple6 "u)ber. T&e" $&e fu"%$#o" f #s "o$ eu!l $o ' s#"%e $&ey &!ve d#ffere"$ do)!#"s.

E-a%5#e .<

*e$ $&e fu"%$#o" f be def#"ed by $&e d#!'r!) 1 7

1 7 3 2

*e$ ! fu"%$#o" ' be def#"ed by $&e for)ul! '6B L 6 7 9&ere $&e do)!#" of ' #s $&e se$ N1 7. T&e" f L ' s#"%e $&ey bo$& &!ve 27

MTH 131


$&e s!)e do)!#" !"d s#"%e f !"d ' !ss#'" $&e s!)e #)!'e $o e!%& ele)e"$ #" $&e do)!#". E-a%5#e .3<

3.6

*e$ f: ℜ → ℜ !"d ': ℜ → ℜ. Suppose f #s def#"ed by f6B L 67 !"d ' by 'yB L y7. T&e" f !"d ' !re eu!l fu"%$#o"s $&!$ #s f L '. o$#%e $&!$ 6 !"d y !re )erely du))y v!r#!ble #" $&e for)ul!s def#"#"' $&e fu"%$#o"s.

Range o0 a F!n(t"on

*e$ f be $&e )!pp#"' of A #"$o 0 $&!$ #s le$ f: A → 0. E!%& ele)e"$ #" 0 "eed "o$ !ppe!r !s $&e #)!'e of !" ele)e"$ #" A. (e def#"e $&e r!"'e of f $o %o"s#s$ pre%#sely of $&ose ele)e"$s #" 0 9&#%& !ppe!r !"d $&e #)!'e of !$ le!s$ o"e ele)e"$ #" A. (e de"o$e $&e r!"'e of f: A → 0 y fAB fAB o$#%e $&!$ fAB #s ! subse$ of 0. #.e fAB E-a%5#e 3.1

*e$ $&e fu"%$#o" f: ℜ → ℜ be def#"ed by $&e for)ul! f6B L 67. T&e" $&e r!"'e of f %o"s#s$s of $&e pos#$#ve re!l "u)bers !"d ero.

E-a%5#e 3.

*e$ f: A → 0 be $&e fu"%$#o" #" E6!)ple 1.3. T&e" fAB L Nb %

3.7

One  One In'e(t")e F!n(t"ons

*e$ f )!p A #"$o 0. T&e" f #s %!lled ! one-one or Injective function #f d#ffere"$ ele)e"$s #" 0 !re !ss#'"ed $o d#ffere"$ ele)e"$s #" A $&!$ #s #f "o $9o d#ffere"$ ele)e"$s #" A &!ve $&e s!)e #)!'e. More br#efly f: A → 0 #s o"e/o"e #f f!B L f!JB #)pl#es ! L !a or eu#v!le"$ly ! L !a #)pl#es f!B ≠ f!aB E-a%5#e 6.1<

*e$ $&e fu"%$#o" f: ℜ → ℜ be def#"ed by $&e for)ul! f6B L 67. T&e" f #s "o$ ! o"e/o"e fu"%$#o" s#"%e f7B L f/7B L 2 $&!$ #s s#"%e $&e #)!'e of 23

MTH 131


$9o d#ffere"$ re!l "u)bers 7 !"d /7 #s $&e s!)e "u)ber 2. E-a%5#e 6.<

*e$ $&e fu"%$#o" f: ℜ → ℜ be def#"ed by $&e for)ul! f6B L 63. T&e" f #s ! o"e/o"e )!pp#"' s#"%e $&e %ubes of $&e d#ffere"$ re!l "u)bers !re $&e)selves d#ffere"$.

E-a%5#e 6.3<

T&e fu"%$#o" f 9&#%& !ss#'"s $o e!%& %ou"$ry #" $&e 9orld #$s %!p#$!l %#$y #s o"e/o"e s#"%e d#ffere"$ %ou"$r#es &!ve d#ffere"$ %!p#$!ls $&!$ #s "o %#$y #s $&e %!p#$!l of $9o d#ffere"$ %ou"$r#es.

3.8

Onto S!&'e(t")e F!n(t"on

*e$ f be ! fu"%$#o" of A #"$o 0. T&e" $&e r!"'e fAB of $&e fu"%$#o" f #s ! subse$ of 0 $&!$ #s fAB ⊂ 0. If fAB L 0 $&!$ #s #f every )e)ber of 0 !ppe!rs !s $&e #)!'e of !$ le!s$ o"e ele)e"$ of A $&e" 9e s!y f #s ! fu"%$#o" of A o"$o 0 or f )!ps A o"$o 0 of f #s !" onto or Subjective function”. E-a%5#e 7.1<

*e$ $&e fu"%$#o" f: ℜ → ℜ be def#"ed by $&e for)ul! f6B L 67. T&e" f #s "o$ !" o"$o fu"%$#o" s#"%e $&e "e'!$#ve "u)bers do "o$ !ppe!r #" $&e r!"'e of f $&!$ #s "o "e'!$#ve "u)ber #s $&e su!re of ! re!l "u)ber.

E-a%5#e 7.<

*e$ f: A → 0 be $&e fu"%$#o" #" E6!)ple 1.3. o$#%e $&!$ fAB L Nb %. S#"%e 0 L N! b % $&e r!"'e of f does "o$ eu!l %o/do)!#" #.e. #s "o$ o"$o.

E-a%5#e 7.3<

*e$ f: A → 0 be $&e fu"%$#o" #" e6!)ple 1.8: o$#%e $&!$ fAB L N6 y  L 0 $&!$ #s $&e r!"'e of f #s eu!l $o $&e %o/do)!#" 0. T&us f )!ps A o"$o 0 #.e. f #s !" o"$o )!pp#"'.

22

MTH 131

3.9


I+ent"t, F!n(t"on

*e$ A be !"y se$. *e$ $&e fu"%$#o" f: A → A be def#"ed b y $&e for)ul! f6B L 6 $&!$ #s le$ f !ss#'" $o e!%& ele)e"$ #" A $&e ele)e"$ #$self. T&e" f #s %!lled $&e #de"$#$y fu"%$#o" or $&e #de"$#$y $r!"sfor)!$#o" o" A. (e de"o$e $&#s fu"%$#o" by 1 or by 1A. 3.:

Constant F!n(t"ons

A fu"%$#o" f of A o"$o 0 #s %!lled ! constant function #f $&e s!)e ele)e"$ of b∈0 #s !ss#'"ed $o every ele)e"$ #" A. I" o$&er 9ords f: A → 0 #s ! %o"s$!"$ fu"%$#o" #f $&e r!"'e of f %o"s#s$s of o"ly o"e ele)e"$. 3.=

Produ%$ u"%$#o"

*e$ : A → 0 !"d ': 0 → C be $9o fu"%$#o"s $&e" $&e produ%$ of fu"%$#o"s f !"d ' #s de"o$ed ' o fB : A → C def#"ed by ' o fB!B L ' f!BB (e %!" "o9 %o)ple$e our d#!'r!): A

ƒ

'

0

C

' o fB E-a%5#e 9.1<

*e$ f: A → 0 !"d ': 0 → C be def#"ed by $&e d#!'r!)s

28

MTH 131


A

0

C

! b %

6 y 

r s $

(e %o)pu$e ' o fB: A → C by #$s def#"#$#o": ' o fB!B ' o fBbB ' o fB%B

≡ 'f!BB L 'yB L $ ≡ 'f!BB L 'B L r ≡ 'f!BB L 'yB L $

o$#%e $&!$ $&e fu"%$#o" ' o fB #s eu#v!le"$ $o follo9#"' $&e !rro9s fro) A $o C #" $&e d#!'r!)s of $&e fu"%$#o"s f !"d '. E-a%5#e 9.<

To e!%& re!l "u)ber le$ f !ss#'" #$s su!re #.e. le$ f6B L 67. To e!%& re!l "u)ber le$ ' !ss#'" $&e "u)ber plus 3 #.e. le$ '6B L 6  3. T&e" ' o fB6B ≡ f'6BB L f63B L 63B 7 L 67  46  = ' o fB6B ≡ 'f6BB L '67B L 67  3

Re%a$* 6.1<

*e$ f: A → 0. T&e" I0 o f L f !"d f o 1A L f $&!$ #s $&e produ%$ of !"y fu"%$#o" !"d #de"$#$y #s . $&e fu"%$#o" #$self.

3.;.1 Asso("at")"t, o0 P$o+!(ts O0 F!n(t"ons

*e$ f: A → 0 ': 0 → C !"d &: % → D. T&e" !s #llus$r!$ed #" #'ure 2/1 9e %!" for) $&e produ%$#o" fu"%$#o" ' o f: A → C !"d $&e" $&e fu"%$#o"

24

MTH 131


& o ' o fB: A → D. f

g

A



0

C

D

' o fB

& o ' o fB #'. 2.1 S#)#l!rly !s #llus$r!$ed #" #'ure 2/7 9e %!" for) $&e produ%$ fu"%$#o" & o ': 0 //^ D !"d $&e" $&e fu"%$#o" & o 'B o f: A → D. B

A

C

D

& o 'B & o ' o fB

#' 2.7 0o$& & ' fB !"d & 'B f !re fu"%$#o" of A #"$o D. A b!s#% $&eore) o" fu"%$#o"s s$!$es $&!$ $&ese fu"%$#o"s !re eu!l. Spe%#f#%!lly o

o

Teo$e% 6.1<

o

o

*e$ f: A → 0 0 → C !"d &: C → D. T&e" & o 'B o f L & o ' o fB

I" v#e9 of T&eore) 2.1 9e %!" 9r#$e 2>

MTH 131


& o ' o f: A → D 9#$&ou$ !"y p!re"$&es#s. 3.1

In)e$se o0 a F!n(t"on

*e$ f be ! fu"%$#o" of A #"$o 0 !"d le$ b ∈ 0. T&e" $&e inverse of b de"o$ed by f /1 bB Co"s#s$ of $&ose ele)e"$s #" A 9&#%& !re )!pped o"$o b $&!$ #s $&ose ele)e"$ #" A 9&#%& &!ve ) !s $&e#r #)!'e. More br#efly #f f: A → 0 $&e" f /1 bB L N6 6 ∈ AF f6B L b o$#%e $&!$ f /1 bB #s !l9!ys ! subse$ of A. (e re!d f /1 !s f #"verse. E-a%5#e :.1<

! b %

*e$ $&e fu"%$#o" f: A ///^ 0 be def#"ed by $&e d#!'r!) 6 y 

T&e" f /1 6B L Nb % s#"%e bo$& b !"d % &!ve 6 !s $&e#r #)!'e po#"$. Also f /1 yB L N! !s o"ly ! #s )!pped #"$o y. T&e #"verse of  f /1 B #s $&e "ull se$ ∅ s#"%e "o ele)e"$ of A #s )!pped #"$o . E-a%5#e :.<

2?

*e$ f: ℜ ///^ ℜ $&e re!l "u)bers be def#"ed by $&e for)ul! f6B L 6 7. T&e" f /1 2B L N7 /7 s#"%e 2 #s $&e #)!'e of bo$& 7 !"d /7 !"d $&ere #s "o o$&er re!l "u)ber 9&ose su!re #s four. o$#%e $&!$ f /1 /3B L ∅ s#"%e $&ere #s "o ele)e"$ #" ℜ 9&ose su!re #s /3.

MTH 131

E-a%5#e :.3<


*e$ f be ! fu"%$#o" of $&e %o)ple6 "u)bers #"$o $&e %o)ple6 "u)bers 9&ere f #s def#"ed by $&e for)ul! f6B L 67. T&e" f /1 3B L N √3i / √3i !s $&e su!re of e!%& of $&ese "u)bers #s /3.

o$#%e $&!$ $&e fu"%$#o" #" E6!)ple ?.7 !"d ?.3 !re d#ffere"$ !l$&ou'& $&ey !re def#"ed by $&e s!)e for)ul! (e "o9 e6$e"d $&e def#"#$#o" of $&e #"verse of ! fu"%$#o". *e$ f: A //^ 0 !"d le$ D be ! subse$ of 0 $&!$ #s D ⊂ 0. T&e" $&e #"verse of D u"der $&e )!pp#"' f de"o$ed by f 1 DB %o"s#s$s of $&ose ele)e"$s #" A 9&#%& !re )!pped o"$o so)e ele)e"$ #" D. More br#efly f /1 DB L N6 6 ∈ A f6B ∈ D E-a%5#e ;.1<

*e$ $&e fu"%$#o" f: A //^ 0 be def#"ed by $&e d#!'r!) 6

r

y

s



$

T&e" f /1 Nr sB L Ny s#"%e o"ly y #s )!pped #"$o r or s. Also f /1 Nr $B L N6 y  L A s#"%e e!%& ele)e"$ #" A !s #$s #)!'e r or $. E-a%5#e ;.<

*e$ f: ℜ ///^ ℜ be def#"ed by f6B L 6 7 !"d le$ D L _2 =` L N6  2 ≤ 6 ≤ = T&e" f /1 DB L N6  /3 ≤ 6 ≤ /7 or 7 ≤ 6 ≤ 3

E-a%5#e ;.3<

*e$ f: A ///^ 0 be !"y fu"%$#o". T&e" f /1 0B L A s#"%e every ele)e"$ #" A &!s #$s #)!'e #" 0. If fAB de"o$e $&e r!"'e of $&e fu"%$#o" f $&e"

f /1 fABB L A 2=

MTH 131


ur$&er #f b ∈ 0 $&e" f /1bB L f /1NbB Here f /1 &!s $9o )e!"#"'s !s $&e #"verse of !" ele)e"$ of 0 !"d !s $&e #"verse of ! subse$ of 0. 3.11 In)e$se F!n(t"on

*e$ f be ! fu"%$#o" of A #"$o 0. I" 'e"er!l f /1bB %ould %o"s#s$ of )ore $&!" o"e ele)e"$ or )#'&$ eve" be e)p$y se$ ∅. o9 #f f: A → 0 #s ! o"e/o"e fu"%$#o" !"d !" o"$o fu"%$#o" $&e" for e!%& b ∈ 0 $&e #"verse f /1 bB 9#ll %o"s#s$ of ! s#"'le ele)e"$ #" A. (e $&erefore &!ve ! rule $&!$ !ss#'"s $o e!%& b ∈ 0 ! u"#ue ele)e"$ f /1bB #" A. A%%ord#"'ly f /1 #s ! fu"%$#o" of 0 #"$o A !"d 9e %!" 9r#$e f /1 : 0 → A I" $&#s s#$u!$#o" 9&e" f: A → 0 #s o"e/o"e !"d o"$o 9e %!ll f /1 $&e #"verse fu"%$#o" of f. E-a%5#e 1.1<

*e$ $&e fu"%$#o" f: A → 0 be def#"ed by $&e d#!'r!) !

f

6

b

y

%



o$#%e $&!$ f #s o"e/o"e !"d o"$o. T&erefore f /1 $&e #"verse fu"%$#o" e6#s$s (e des%r#be f /1: 0 → A by $&e d#!'r!)

6 8<

f =1

!

MTH 131


E-a%5#e 8.1<

y

b



%

*e$ $&e fu"%$#o" f be def#"ed by $&e d#!'r!): !

1

b

7

%

3

T&e" f #s ! %o"s$!"$ fu"%$#o" s#"%e 3 #s !ss#'"ed $o every ele)e"$ #" A. E-a%5#e 8.3<

3.;

*e$ f: ℜ → ℜ be def#"ed by $&e for)ul! f6B L 8. T&e" f #s ! %o"s$!"$ fu"%$#o" s#"%e 8 #s !ss#'"ed $o every ele)e"$.

P$o+!(t F!n(t"on

*e$ f be ! fu"%$#o" of A !"d 0 !"d le$ ' be ! fu"%$#o" of 0 $&e %o/ do)!#" of f #"$o C. (e #llus$r!$e $&e fu"%$#o" belo9.

A

f

0

g

C

*e$ ! ∈ AF $&e" #$s #)!'e f 6B #s #" 0 9&#%& #s $&e do)!#" of '. A%%ord#"'ly 9e %!" f#"d $&e #)!'e of f !B u"der $&e )!pp#"' of ' $&!$ #s 9e %!" f#"d ' f!BB. T&us 9e &!ve ! rule 9&#%& !ss#'"s $o e!%& ele)e"$ ! ∈A ! %orrespo"d#"' ele)e"$ f!BB ∈ C. I" o$&er 9ords 9e &!ve ! fu"%$#o" of A #"$o C. T&#s "e9 fu"%$#o" #s %!lled $&e product function or composition function of f !"d ' !"d #$ #s de"o$ed by 81

MTH 131


' o fB or 'fB More br#efly #f f: A → 0 !"d ': 0 → C $&e" 9e def#"e ! fu"%$#o" o$#%e fur$&er $&!$ #f 9e se"d $&e !rro9s #" $&e oppos#$e d#re%$#o" #" $&e f#rs$ d#!'r!) of f 9e esse"$#!lly &!ve $&e d#!'r!) of f /1. E-a%5#e 1.<

*e$ $&e fu"%$#o" f: A → 0 be def#"ed by $&e d#!'r!) !

6

b %

y 

S#"%e f!B L y !"d f%B L y $&e fu"%$#o" f #s "o$ o"e/o"e. T&erefore $&e #"verse fu"%$#o" f /1 does "o$ e6#s$. As f /1 yB L N! % 9e %!""o$ !ss#'" bo$& ! !"d % $o $&e ele)e"$ y ∈ 0. E-a%5#e 1.3<

3.11.1

*e$ f: ℜ → ℜ $&e re!l "u)bers be def#"ed by f6B L 63. o$#%e $&!$ f #s o"e/o"e !"d o"$o. He"%e f /1: ℜ → ℜ e6#s$s. I" f!%$ 9e &!ve ! for)ul! 9&#%& def#"es $&e #"verse fu"%$#o" f /1 6B L 3√6.

Teo$e%s on te "n)e$se F!n(t"on

*e$ ! fu"%$#o" f: A → 0 &!ve !" #"verse fu"%$#o" f /1: 0 → A. T&e" 9e see by $&e d#!'r!)

87

MTH 131

E*EMETAR SET THEOR f -!

f A

A

B

" f -! o 0

T&!$ 9e %!" for) $&e produ%$ f /1 o fB 9&#%& )!ps A #"$o A !"d 9e see by $&e d#!'r!) f -!

f

B

A

" f

o

B

0 -! #

T&!$ 9e %!" for) $&e produ%$ fu"%$#o" f o f /1B 9&#%& )!ps 0 #"$o 0. (e "o9 s$!$e $&e b!s#% $&eore)s o" $&e #"verse fu"%$#o": Teo$e% 6.<

*e$ $&e fu"%$#o" f: A → 0 be o"e/o"e !"d o"$oF #.e. $&e #"verse fu"%$#o" f /1: 0 → A e6#s$s. T&e" $&e produ%$ fu"%$#o" f /1 o fB: A → A

#s $&e #de"$#$y fu"%$#o" o" A !"d $&e produ%$ fu"%$#o" f o f /1B: 0 → 0 #s $&e #de"$#$y fu"%$#o" o" 0. Teo$e% 6.3<

*e$ f: A → 0 !"d ': 0 → A. T&e" ' #s $&e #"verse fu"%$#o" of f #.e. ' L f  1 #f $&e produ%$ fu"%$#o" ' o fB: A → A #s $&e 83

MTH 131


#de"$#$y fu"%$#o" o" A !"d f o 'B: 0 → 0 #s $&e #de"$#$y fu"%$#o" o" 0. 0o$& %o"d#$#o"s !re "e%ess!ry #" T&eore) 2.3 !s 9e s&!ll see fro) $&e e6!)ple belo9 6 y

!

!

b

b

%

%

!B

6 y bB

o9 def#"e ! fu"%$#o" ': 0 → A by $&e d#!'r!) bB !bove. (e %o)pu$e ' o fB: A → A ' o fB6B L ' f6BB L '%B L 6 !"d ' o fByB L ' fyBB L '!B L y T&erefore $&e produ%$ fu"%$#o" ' o fB #s $&e #de"$#$y fu"%$#o" o" A. 0u$ ' #s "o$ $&e #"verse fu"%$#o" of f be%!use $&e produ%$ fu"%$#o" f o GB #s "o$ $&e #de"$#$y fu"%$#o" o" 0 f "o$ be#"' !" o"$o fu"%$#o". 6.

CONCLUSION

I bel#eve $&!$ by "o9 you fully 'r!sp $&e #de! of fu"%$#o"s )!pp#"'s !"d $r!"sfor)!$#o"s. T&#s +"o9led'e 9#ll be bu#l$ upo" #" subseue"$ u"#$s. 7.

SUMMARY

Re%!ll $&!$ #" $&#s u"#$ 9e &!ve s$ud#ed %o"%ep$s su%& !s )!pp#"'s !"d fu"%$#o"s. (e &!ve !lso e6!)#"ed $&e %o"%ep$s of o"e/$o/o"e !"d o"$o fu"%$#o"s. T&#s %o"%ep$ &!s !llo9ed us $o e6pl!#" eu!l#$y be$9ee" $9o se$s. (e !lso es$!bl#s&ed #" $&e u"#$ $&!$ $&e #"verse of f: A  0 usu!lly de"o$ed f /1 e6#$ #f f #s ! o"e/$o/o"e !"d o"$o fu"%$#o". 82

MTH 131


I$ #s #"s$ru%$#ve $o "o$e $&!$ I"verse fu"%$#o" #s "o$ s$ud#ed #" #sol!$#o" bu$ )ore #)por$!"$ly ! useful !"d po9erful $ool #" u"ders$!"d#"' %!l%ulus. 8.

TUTOR  MAR?ED ASSIGNMENTS

1.

7.

3. 2. 9.

*e$ $&e fu"%$#o" f: R  → R  be def#"ed by f B L N1 #f 6 #s r!$#o"!l` N/1 #f 6 #s #rr!$#o"!l. !. E6press f #" 9ords b. Suppose $&e ordered p!#rs 6  y 1B !"d 3 6  yB !re eu!l. #"d 6 !"d y. *e$ M L N1 7 3 2 8 !"d le$ $&e fu"%$#o" f: M → ℜ be def#"ed by f6B L 67  76 /1 #"d $&e 'r!p& of f. Prove: A 6 0 ∩ CB L A 6 0B ∩ A 6 CB Prove A ⊂ 0 !"d C ⊂ D #)pl#es A 6 CB ⊂ 0 6 DB.



88

MTH 131

UNIT 7


FUNCTIONS II

CONTENTS

1.< 7.< 3.<

2.< 8.< 4.< >.<

I"$rodu%$#o" Ob,e%$#ves M!#" 0ody 3.1 Ordered P!#rs 3.7 Produ%$ Se$ 3.3 Coord#"!$e D#!'r!)s 3.2 Gr!p& of ! u"%$#o" 3.2.1 Proper$#es of $&e 'r!p& of ! fu"%$#o" 3.8 Gr!p&s !"d Coord#"!$e d#!'r!)s 3.8.1 Proper$#es of Gr!p&s of u"%$#o"s o" Coord#"!$e d#!'r!)s 3.4 u"%$#o"s !s se$s of ordered p!#rs 3.> Produ%$ Se$s #" Ge"er!l Co"%lus#o" Su))!ry Tu$or M!r+ed Ass#'")e"$s Refere"%es !"d ur$&er Re!d#"'s.

1.

INTRODUCTION

I" $&#s u"#$ 9e !re 'o#"' $o def#"e ! $ype of se$ $&!$ "o$ o"ly '#ves ! be$$er u"ders$!"d#"' of C!r$es#!" %oord#"!$e bu$ !lso br#"'s $&e %o"%ep$ of re!l/v!lued fu"%$#o"s $o $&e fore. .

O4ECTIVES

A$ $&e e"d of $&#s u"#$ you s&ould be !ble $o:    

84

#"d $&e ordered p!#rs '#ve" $9o se$s #"d $&e ordered p!#rs %orrespo"d#"' $o $&e po#"$s o" $&e C!r$es#!" %oord#"!$e d#!'r!) #"d $&e 'r!p& of fu"%$#o"s S$!$e 9&e$&er or "o$ ! se$ of ordered p!#rs of ! '#ve" se$ s!y A #s ! fu"%$#o" of A #"$o #$self.

MTH 131


3.

MAIN ODY

3.1

O$+e$e+ Pa"$s

I"$u#$#vely !" ordered pair %o"s#s$s of $9o ele)e"$s s!y ! !"d b #" 9&#%& o"e of $&e) s!y ! #s des#'"!$ed !s $&e f#rs$ ele)e"$ !"d $&e o$&er !s $&e se%o"d ele)e"$. A" ordered p!#r #s de"o$ed by ! bB T9o ordered p!#rs ! bB !"d % dB !re eu!l #f !"d o"ly #f ! L % !"d b L d. E-a%5#e 1.1< E-a%5#e 1.< E-a%5#e 1.3< E-a%5#e 1.6<

T&e ordered p!#rs 7 3B !"d 3 7B !re d#ffere"$ T&e po#"$s #" $&e C!r$es#!" pl!"e s&o9" #" f#' 8.1 belo9 represe"$ ordered p!#rs of re!l "u)bers. T&e se$ N7 3 #s "o$ !" ordered p!#r s#"%e $&e ele)e"$s 7 !"d 3 !re "o$ d#s$#"'u#s&ed Ordered p!#rs %!" &!ve $&e s!)e f#rs$ !"d se%o"d ele)e"$s su%& !s 1 1B 2 2B !"d 8 8B.

Al$&ou'& $&e "o$!$#o" ! bB #s !lso used $o de"o$e !" ope" #"$erv!l $&e %orre%$ )e!"#"' 9#ll be %le!r fro) $&e %o"$e6$. Re%a$* 7.1< A" ordered p!#r ! bB %!" be def#"ed r#'orously by

! bB L N N! N! b  ro) $&#s def#"#$#o" $&e fu"d!)e"$!l proper$y of ordered p!#rs %!" be prove": ! bB L % dB #)pl#es ! L % !"d b L d 3.

P$o+!(t Set

*e$ A !"d 0 be $9o se$s. T&e product set of A !"d 0 %o"s#s$s of !ll ordered p!#rs ! bB 9&ere ! ∈A !"d b∈0. #$ #s de"o$ed by A 6 0. (&#%& re!ds A %ross 0. More pre%#sely 8>

MTH 131


A 6 0 L N ! bB  !∈A b∈0 E-a%5#e .1<

*e$ A L N1 7 3 !"d 0 L N! b. T&e" $&e produ%$ se$ A 6 0 L N1 !B 1 bB 7 !B 7 bB 3 !B 3 bB

E-a%5#e .<

*e$ ( L Ns $. T&e" ( 6 ( L Ns sB s $B $ sB $ $B

E-a%5#e .3<

T&e C!r$es#!" pl!"e s&o9" #" #' 8.1 #s $&e produ%$ se$ of $&e re!l "u)bers 9#$& #$self #.e. ℜ 6 ℜ

T&e produ%$ se$ A 6 0 #s !lso %!lled $&e $artesian %roduct of A !"d 0. #$ #s "!)ed !f$er $&e )!$&e)!$#%#!" Des%!r$es 9&o #" $&e seve"$ee"$& %e"$ury f#rs$ #"ves$#'!$ed $&e se$ ℜ 6 ℜ. I$ #s !lso for $&#s re!so" $&!$ ℜ 6 ℜ !s p#%$ured #" #'. 8.1 #s %!lled $&e C!r$es#!" Pl!"e. Re%a$* 7.< If se$ A &!s " ele)e"$s !"d se$ 0 &!s ) ele)e"$s $&e" $&e produ%$ se$ A 6 0 &!s n times m ele)e"$s #.e. nm

ele)e"$s. If e#$&er A or 0 #s $&e "ull se$ $&e" A 6 0 #s !lso $&e "ull se$. *!s$ly #f e#$&er A or 0 #s #"f#"#$e !"d $&e o$&er #s "o$ e)p$y $&e" ! 6 0 #s #"f#"#$e.

Re%a$* 7.3< T&e

C!r$es#!" produ%$ of $9o se$s #s "o$ %o))u$!$#veF )ore spe%#f#%!lly A 6 0 ≠ 0 6 A U"less A L 0 or o"e of $&e f!%$ors #s e)p$y.

3.3

Coo$+"nate D"ag$a%s

ou !re f!)#l#!r 9#$& $&e C!r$es#!" pl!"e ℜ 6 ℜ !s s&o9" #" #' 8.1 belo9. E!%& po#"$ P represe"$s !" ordered p!#r ! bB of re!l "u)bers. A ver$#%!l $&rou'& P )ee$s $&e &or#o"$!l !6#s !$ a !"d ! &or#o"$!l l#"e $&rou'& P )ee$s $&e ver$#%!l !6#s !$ b !s #" #'. 8.1.

8?

MTH 131


#'. 8.1 T&e C!r$es#!" produ%$ of !"y $9o se$s #f $&ey do "o$ %o"$!#" $oo )!"y1 ele)e"$s %!" be d#spl!yed o" ! %oord#"!$e d#!'r!) #" ! s#)#l!r )!""er. or e6!)ple #f A L N! b % d !"d 0 L N6 y  $&e" $&e %oord#"!$e d#!'r!) of A 6 0 #s !s s&o9" #" #' 8.7 belo9. Here $&e ele)e"$s of A !re d#spl!yed o" $&e &or#o"$!l !6#s !"d $&e ele)e"$s of 0 !re d#spl!yed o" $&e ver$#%!l !6#s. o$#%e $&!$ $&e ver$#%!l l#"es $&rou'& $&e ele)e"$s of A !"d $&e &or#o"$!l l#"es $&rou'& $&e ele)e"$s of 0 )ee$ 17 po#"$s. T&ese po#"$s represe"$ A 6 0 #" $&e obv#ous 9!y. T&e po#"$ P #s $&e ordered p!#r % yB.

8=

MTH 131

3.6


G$a5 o0 A F!n(t"on

*e$ f be ! fu"%$#o" of A #"$o 0 $&!$ #s le$ f: A ///^ 0. T&e 'r!p& fc of $&e fu"%$#o" f %o"s#s$s of !ll ordered p!#rs #" 9&#%& ! ∈A !ppe!rs !s ! f#rs$ ele)e"$ !"d #$s #)!'e !ppe!rs !s #$s se%o"d ele)e"$. I" o$&er 9ords fc L N! bB  !∈A b  f!Bs o$#%e $&!$ fc $&e 'r!p& of f: A //^ 0 #s ! subse$ of A 6 0. E-a%5#e 3.1<

*e$ $&e fu"%$#o" f: A //^ 0 be def#"ed by $&e d#!'r!) A

0

! 1

b %

7

d

3

T&e" f!B L 7 fbB L 3 f%B L 7 !"d fdB L 1. He"%e $&e 'r!p& of f #s c L N! 7B b 3B % 7B d 1B E-a%5#e 3.<

*e$ ( L N1732. *e$ $&e fu"%$#o" f: ( //^ ℜ be def#"ed by f6B L 6  3 T&e" $&e 'r!p& of f #s fc L N1 2B 7 8B 3 4B 2 >B

E-a%5#e 3.3<

4<

*e$  be $&e "!$ur!l "u)bers 1 7 3@@@*e$ $&e fu"%$#o" ':

MTH 131


 →  be def#"ed by '6B L 63 T&e" $&e 'r!p& of ' #s 'c L N11B 7?B 3 7>B 2 42B@.. 3.6.1 P$o5e$t"es o0 te G$a5 o0 a 0!n(t"on

*e$ f: A → 0. (e re%!ll $9o proper$#es of $&e fu"%$#o" f. #rs$ for e!%& ele)e"$ !∈A $&ere #s !ss#'"ed !" ele)e"$ #" 0. Se%o"dly $&ere #s o"ly o"e ele)e"$ 0 9&#%& #s !ss#'"ed $o e!%& ! ∈A. I" v#e9 of $&ese proper$#es of f $&e 'r!p& fc of f &!s $&e follo9#"' $9o proper$#es: P$o5e$t, 1<

or e!%& !∈A $&ere #s !" ordered p!#r ! bB ∈ fc

E!%& !∈A !ppe!rs !s $&e f#rs$ ele)e"$ #" o"ly o"e ordered p!#r #" fc $&!$ #s ! bB ∈ fc ! %B ∈ fc #)pl#es b L % P$o5e$t, <

I" $&e follo9#"' e6!)ples le$ A L N1732 !"d 0 L N3284 E-a%5#e 6.1<

T&e se$ of ordered p!#rs N18B 73B 24B

%!""o$ be $&e 'r!p& of ! fu"%$#o" of A #"$o 0 s#"%e #$ v#ol!$es proper$y 1. Spe%#f#%!lly 3∈A !"d $&ere #s "o ordered p!#r #" 9&#%& 3 #s ! f#rs$ ele)e"$. E-a%5#e 6.<

T&e se$ of ordered p!#rs N18B 73B 34B 24B 72B.

%!""o$ be $&e 'r!p& of ! fu"%$#o" of A #"$o 0 s#"%e #$ v#ol!$es Proper$y 7 $&!$ #s $&e ele)e"$ 7 ∈A !ppe!rs !s $&e f#rs$ ele)e"$ #" $9o d#ffere"$ ordered p!#rs 7 3B !"d 72B 41

MTH 131

3.7


G$a5s an+ Coo$+"nate D"ag$a%s

*e$ fc be $&e 'r!p& of ! fu"%$#o" f: A → 0. As fc #s ! subse$ of A 6 0 #$ %!" be d#spl!yed #.e. 'r!p&ed o" $&e %oord#"!$e d#!'r!) of A 6 0. E-a%5#e 7.1<

*e$ f6B L 67 def#"e ! fu"%$#o" o" $&e #"$erv!l / 7≤ 6 ≤ 2. T&e" $&e 'r!p& of f #s d#spl!yed #" #' 8.3 belo9 #" $&e usu!l 9!y: 18 1< 8 /7

/1

/8

<

1

7

3

2

#' 8.3 E-a%5#e 7.<

*e$ ! fu"%$#o" f: A → 0 be def#"ed by $&e d#!'r!) s&o9" #" #' 8.2 belo9 Here fc $&e 'r!p& of f %o"s#s$ of $&e ordered p!#rs ! 7B b 3B % 1B !"d d 7B. T&e" fc #s d#spl!yed o" $&e %oord#"!$e d#!'r!) A 6 0 !s s&o9" #" #' 8.8 belo9.

47

MTH 131


A

0

!

1

b

7

%

3

d #'. 8.2

3 7 1 !

b

%

#'. 8.8

3.7.1 P$o5e$t"es o0 G$a5s o0 F!n(t"ons on Coo$+"nate D"ag$a%s

*e$ f: A → 0. T&e" fc $&e 'r!p& of f &!s $&e $9o proper$#es l#s$ed prev#ously: P$o5e$t, 1< P$o5e$t, <

or e!%& ! ∈A $&ere #s !" ordered p!#r ! bB ∈ fc If ! bB ∈ fc !"d ! %B ∈ fc $&e" b L %.

T&erefore #f fc #s d#spl!yed o" $&e %oord#"!$e d#!'r!) of A 6 0 #$ &!s $&e follo9#"' $9o proper$#es: P$o5e$t, 1< P$o5e$t, <

E!%& ver$#%!l l#"e 9#ll %o"$!#" !$ le!s$ o"e po#"$ of fc E!%& ver$#%!l l#"e 9#ll %o"$!#" o"ly o"e po#"$ of fc

E-a%5#e 8.1<*e$

! L N! b % !"d 0 L N1 7 3. Co"s#der $&e se$s of po#"$s #" $&e $9o %oord#"!$e d#!'r!)s of A 6 0 belo9.

43

d

MTH 131


I" 1B $&e ver$#%!l l#"e $&rou'& b does "o$ %o"$!#" ! po#"$ of $&e se$F &e"%e $&e se$ of po#"$s %!""o$ be $&e 'r!p& of ! fu"%$#o" of A #"$o 0. I" 7B $&e ver$#%!l l#"e $&rou'& a %o"$!#"s $9o po#"$s of $&e se$ &e"%e $&#s se$ of po#"$ %!""o$ be $&e 'r!p& of ! fu"%$#o" of A #"$o 0. T&e %#r%le 67  y7 L = p#%$ured belo9 %!""o$ be $&e 'r!p& of ! fu"%$#o" s#"%e $&ere !re ver$#%!l l#"es 9&#%& %o"$!#" )ore $&!" o"e po#"$ of $&e %#r%le.

E-a%5#e 8.<

/2

/7

7

2

/7

/2

67  y7 L = #s plo$$ed 3.8

F!n(t"ons as Sets o0 O$+e$e+ Pa"$s

*e$ fc be ! subse$ of A 6 0 $&e C!r$es#!" produ%$ of se$s A !"d 0F !"d le$ fc &!ve $&e $9o proper$#es d#s%ussed prev#ously: P$o5e$t, 1< P$o5e$t, <

42

or e!%& ! ∈ A $&ere #s !" ordered p!#r ! bB ∈fc. o $9o d#ffere"$ ordered p!#rs #" fc &!ve $&e s!)e f#rs$ ele)e"$.

MTH 131


T&us 9e &!ve ! rule $&!$ !ss#'"s $o e!%& ele)e"$ ! ∈ A $&e ele)e"$ b ∈ 0 $&!$ !ppe!r #" $&e ordered p!#r ! bB ∈ fc. Proper$y 1 'u!r!"$ees $&!$ e!%& ele)e"$ #" A 9#ll &!ve !" #)!'e !"d Proper$y 7 'u!r!"$ees $&!$ $&e #)!'e #s u"#ue. A%%ord#"'ly fc #s ! fu"%$#o" of A #"$o 0. I" v#e9 of $&e %orrespo"de"%e be$9ee" fu"%$#o"s f: A → 0 !"d subse$ of A 6 0 9#$& proper$y 1 !"d proper$y 7 !bove 9e redef#"e ! fu"%$#o" by $&e De0"n"t"on 7.1<

A fu"%$#o" f of A #"$o 0 #s ! subse$ of A 6 0 #" 9&#%& e!%& ! ∈ A !ppe!rs !s $&e f#rs$ ele)e"$ #" o"e !"d o"ly o"e ordered p!#r belo"'#"' $o f.

Al$&ou'& $&#s def#"#$#o" of ! fu"%$#o" )!y see) !r$#f#%#!l #$ &!s $&e !dv!"$!'e $&!$ #$ does "o$ use su%& u"def#"ed $er)s !s !ss#'"s rules %orrespo"de"%e. E-a%5#e 9.1<

*e$ A L ! b %B !"d 0 L 1 7 3B. ur$&er)ore le$ f L N! 7B % 1B b 7B

T&e" f &!s Proper$y 1 !"d Proper$y 7. He"%e f #s ! fu"%$#o" of A #"$o 0 9&#%& #s !lso #llus$r!$ed #" $&e follo9#"' d#!'r!): A 0

E-a%5#e 9.<

!

1

b

7

%

3

*e$ 5 L N1 7 3 !"d ( L N! e I o u. Also le$ f  N1 !B 7 eB 3 1B 7 uB T&e" f #s "o$ ! fu"%$#o" of 5 #"$o ( s#"%e $9o d#ffere"$ ordered p!#rs #" f 7 eB !"d 7 uB &!ve $&e s!)e f#rs$ ele)e"$. If f #s $o be ! fu"%$#o" of 5 #"$o ( $&e" #$ %!""o$ !ss#'" bo$& e !"d u $o $&e ele)e"$ 7 ∈5. 48

MTH 131


E-a%5#e 9.3<

*e$ S L N1732 !"d T L N138. *e$ f L N11B 7 8B 2 3B T&e" f #s "o$ ! fu"%$#o" of S #"$o T s#"%e 3 ∈S does "o$ !ppe!r !s $&e f#rs$ ele)e"$ #" !"y ordered p!#r belo"'#"' $o f.

T&e 'eo)e$r#%!l #)pl#%!$#o" of Def#"#$#o" 8.1 #s s$!$ed #". Re%a$* 7.6<

*e$ f be $&e se$ of po#"$s #" $&e %oord#"!$e d#!'r!) of A 6 0. #f every ver$#%!l l#"e %o"$!#"s o"e !"d o"ly po#"$ of f $&e" f #s ! fu"%$#o" of A #"$o 0.

Re%a$* 7.7<

*e$ $&e fu"%$#o" f: A → 0 be o"e/o"e !"d o"$o. T&e" $&e #"verse fu"%$#o" f 1 %o"s#s$s of $&ose ordered p!#rs 9&#%& 9&e" reversed #.e. per)u$ed belo"' $o f. More spe%#f#%!lly f 1 L Nb !B  ! bB ∈ f

3.9

P$o+!(t Sets "n Gene$a#

T&e %o"%ep$ of ! produ%$ se$ %!" be e6$e"ded $o )ore $&!" $9o se$s #" ! "!$ur!l 9!y. T&e C!r$es#!" produ%$ of se$s A 0 !"d C de"o$ed by A 606C Co"s#s$s of !ll ordered $r#ple$s ! b %B 9&ere ! ∈A b∈0 !"d %∈C. A"!lo'ously $&e C!r$es#!" produ%$ of " se$s A 1 A7@@A" de"o$ed by A1 6 A7 6@ 6 A " Co"s#s$s of !ll ordered "/$uples ! 1 !7@!"B 9&ere !1∈A@. !"∈A. Here !" ordered "/$uple &!s $&e obv#ous #"$u#$#ve )e!"#"' $&!$ #s #$ %o"s#s$s of " ele)e"$s "o$ "e%ess!r#ly d#s$#"%$ #" 9&#%& o"e of $&e) #s des#'"!$ed !s $&e f#rs$ ele)e"$ !"o$&er !s $&e se%o"d ele)e"$ e$%. E-a%5#e :.1<

44

I" $&ree/d#)e"s#o"!l Eu%l#de!" 'eo)e$ry e!%& po#"$ represe"$s !" ordered $r#ple$ #.e. #$s x/

MTH 131


%o)po"e"$ #$s %o)po"e"$. E-a%5#e :.<

y/%o)po"e"$

!"d #$s

z



*e$ A L N! b 0 L N1 7 3 !"d C L N6 y. T&e" A 6 0 6 C L N! 1 6B ! 1 yB ! 7 6B ! 7 yB ! 3 6B ! 3 yB b 1 6B b 1 yB b 7 6B b 7 yB b 3 6B b 3 yB

6.

CONCLUSION

I" $&#s u"#$ you &!ve s$ud#ed %o"%ep$s su%& !s ordered p!#rs produ%$ se$s %o/ord#"!$e d#!'r!) u"%$#o"s !s se$ of ordered p!#rs. (e &!ve !lso le!r"$ !bou$ &o9 $o represe"$ fu"%$#o" o" ! 'r!p&. (e reu#re $&e )!s$ery of $&e !bove %o"%ep$s #" $&e u"ders$!"d#"' of $&e subseue"$ u"#$s. 7.

SUMMARY

Re%!ll $&e follo9#"': T&!$ !" ordered p!#r #s de"o$ed by ! 0B ! ∈ A !"d b ∈ 0. T9o ordered p!#rs ! bB !"d % dB !re #f !"d o"ly #s ! L % !"d b L d. T&!$ #s A !"d 0 !re $9o se$s su%& $&!$ ! ∈ A !"d b ∈ 0 $&e) $&e produ%$ of A !"d 0 #s de"o$ed by A  0 L N! bB ! ∈ A b ∈ 0 T&!$ $&e C!r$es#!" pl!%e #s $&e produ%$ se$ of re!l "u)ber 9#$& #$self #.e IR  IR T&!$ $&e %o"%ep$ of produ%$ %!" be e6$e"ded $o )ore $&!" $9o se$s #" ! "!$ur!l 9!ys #.e #f A 0 !"d C !re se$s $&e" $&e produ%$ of A 0 !"d C #s de"o$ed !s A  0  C L N! b %B ! ∈ A b ∈ 0 % C Ge"er!lly $&e C!r$es#!" produ%$ of " se$s A 1 A7 @@..A" #s de"o$ed by 4>

MTH 131


A1 A7 A3 @.  A" L N!1 !7 !3 @@!"B !1 A1 !7 A7 @@@ !" ∈ A"  8.


1.

9.

7.

Suppose $&e ordered p!#rs 6  y 1B !"d 3 6  yB !re eu!l. #"d 6 !"d y. *e$ M L N17328 !"d le$ $&e fu"%$#o" f: M → ℜ be def#"ed by f6B L 67  76  1 #"d $&e 'r!p& of f.

3. 2.

Prove: A 6 0 ∩ CB L  A 6 0B ∩ A 6 CB Prove A ⊂ 0 !"d C ⊂ D #)pl#es A 6 CB ⊂ 0 6 DB.



Mo+!#e 

4?

MTH 131


U"#$ 1 U"#$ 7 U"#$ 3

UNIT 1

Rel!$#o"s ur$&er T&eory of Se$s ur$&er T&eory of u"%$#o"s Oper!$#o"

RELATIONS

CONTENTS

1.< 7.< 3.<

2.< 8.< 4.< >.< 1.

I"$rodu%$#o" Ob,e%$#ves M!#" body 3.1 Propos#$#o"!l u"%$#o"s Ope" Se"$e"%es 3.7 Rel!$#o"s 3.3 Solu$#o" Se$s !"d Gr!p&s of rel!$#o"s 3.2 Rel!$#o"s !s Se$s of Ordered P!#rs 3.8 Refle6#ve Rel!$#o"s 3.4 Sy))e$r#% Rel!$#o"s 3.> A"$#/Sy))e$r#% Rel!$#o"s 3.? Tr!"s#$#ve Rel!$#o"s 3.= Eu#v!le"%e Rel!$#o"s 3.1< Do)!#" !"d R!"'e of ! Rel!$#o" 3.11 Rel!$#o"s !"d u"%$#o"s Co"%lus#o" Su))!ry Tu$or/M!r+ed !ss#'")e"$s Refere"%es !"d ur$&er Re!d#"'s INTRODUCTION

ro) $&e %o"%ep$ of ordered p!#rs produ%$ se$ or C!r$es#!" produ%$ 9e %!" dr!9 rel!$#o"s b!sed o" propos#$#o"!l fu"%$#o"s def#"ed o" $&e C!r$es#!" produ%$ of $9o se$s T&#s #s 9&!$ 9#ll be developed #" $&#s u"#$ .

O4ECTIVES

4=

MTH 131


Af$er 'o#"' $&rou'& $&#s u"#$ you s&ould be !ble $o do $&e follo9#"': • Der#ve rel!$#o"s !s ordered p!#rs be$9ee" $9o se$s b!sed o" ope"

se"$e"%es. • #"d $&e Do)!#" R!"'e !"d I"verse of ! rel!$#o" • Def#"e 9#$& e6!)ples $&e d#ffere"$ +#"ds of rel!$#o"s • S$!$e 9&e$&er or "o$ ! rel!$#o" def#"ed o" ! se$ #s ! fu"%$#o" of $&e se$ #"$o #$self 3.

MAIN ODY

3.1

P$o5os"t"ona# F!n(t"onsB O5en Senten(es

A P$o5os"t"ona# 0!n(t"on def#"ed o" $&e C!r$es#!" produ%$ A 6 0 of $9o se$s A !"d 0 #s !" e6press#o" de"o$ed by P6yB (&#%& &!s $&e proper$y $&!$ P!bB 9&ere ! !"d b !re subs$#$u$ed for $&e v!r#!bles 6 !"d y respe%$#vely #" P6yB #s $rue or f!lse for !"y ordered p!#r !bB ∈A 6 0. or e6!)ple #f A #s $&e se$ of pl!y9r#'&$ !"d 0 #s $&e se$ of pl!ys $&e" P6yB L 6 9ro$e y Is ! propos#$#o"!l fu"%$#o" o" A 6 0 I" p!r$#%ul!r P S&!+espe!re H!)le$B L S&!+espe!re 9ro$e H!)le$ PS&!+espe!re T&#"'s !ll Ap!r$B L S&!+espe!re 9ro$e T&#"'s !ll Ap!r$ Are $rue !"d f!lse respe%$#vely. T&e e6press#o" P6yB by #$self s&!ll be %!lled !" ope" se"$e"%e #" $9o v!r#!bles or s#)ply !" ope" se"$e"%e. O$&er e6!)ples of ope" se"$e"%es !re !s follo9s: E-a%5#e 1.1<

><

6 #s less $&!" y

MTH 131


6 9e#'&s y +#lo'r!)s 6 d#v#des y 6 #s 9#fe of y T&e su!re of 6 plus $&e su!re of y #s s#6$ee" #.e 67  y7 L 14 E-a%5#e 1.8< Tr#!"'le 6 #s s#)#l!r $o $r#!"'le y E-a%5#e 1.< E-a%5#e 1.3< E-a%5#e 1.6< E-a%5#e 1.7<

I" !ll of our e6!)ples $&ere !re $9o v!r#!ble. I$ #s !lso poss#ble $o &!ve ope" se"$e"%es #" o"e v!r#!ble su%& !s 6 #s #" $&e U"#$ed !$#o"s or #" )ore $&!" $9o v!r#!bles su%& !s 6 $#)es y eu!ls  3.

Re#at"ons

A relation R %o"s#s$s of $&e follo9#"' 1. 7. 3

! se$ A ! se$ 0 !" ope" se"$e"%e P6yB #" 9&%& P! bB #s e#$&er $rue or f!lse for !"y ordered p!#r !bB belo"'#"' $o A 6 0

(e $&e" %!ll R ! relation from A to & !"d de"o$e #$ by R L A 0 P6yBB ur$&er)ore #f !bB #s $rue 9e 9r#$e !Rb 9&#%& re!ds ! #s rel!$ed $o 0. O" $&e o$&er &!"d #f P!bB #s "o$ $rue 9e 9r#$e !R b 9&#%& re!ds ! #s "o$ rel!$ed $o b L  ℜℜ P6yBB 9&ere P6yB re!ds 6 #s less $&!" y. T&e" R 1 #s ! rel!$#o" s#"%e P!bB #.e ! < b #s e#$&er $rue or f!lse for !"y ordered p!#r !bB of re!l "u)bers. Moreover s#"%e P7 πB #s $rue 9e %!" 9r#$e

E-a%5#e .1: *e$ R 1

7 R 1 π !"d s#"%e p8 √7B #s f!lse 9e %!" 9r#$e >1

MTH 131


8 R 1 √7 E-a%5#e <

*e$ R 7 L A0 P6yBB 9&ere A #s $&e se$ of )e" 0 #s $&e se$ of 9o)e" !"d P6yB re!ds 6 #s $&e &usb!"d of y. $&e" R 7 #s ! rel!$#o"

E-a%5#e <3

*e$ R 3 L   P6yBB 9&ere  #s $&e "!$ur!l "u)bers !"d P6y P6yBB re!ds re!ds 6 d#v#de d#v#dess y. T&e" T&e" R 3 #s ! rel!$#o". ur$&er)ore 3 R 3 17 7 R 3 > 8 R 3 18 4 R 2 13

E-a%5#e <6

*e$ R 2 L A 0 P6yBB 9&ere A #s $&e se$ of )e" 0 #s $&e se$ of of 9o)e" !"d !"d P6yB P6yB re!ds 6 d#v#des d#v#des y. y. T&e" R2 #s "o$ ! rel!$#o" s#"%e P!bB &!s "o )e!"#"' #f ! #s ! )!" !"d b #s ! 9o)!".

E-a%5#e <7

*e$ R 8 L   P6yBB 9&ere  #s $&e "!$ur!l "u)bers !"d P6yB re!ds 6 #s less $&!" y. T&e" R 8 #s ! rel!$#o". o$#%e $&!$ R 1 !"d R 8 !re "o$ $&e s!)e rel!$#o" eve" $&ou'& $&e s!)e ope" se"$e"%e #s used $o def#"e e!%& rel!$#o" *e$ R L A 0 P6yBB be ! rel!$#o". (e $&e" s!y $&!$ $&!$ $&e ope" ope" se"$e" se"$e"%e %e P6yB defines a relation fro) A $o 0. ur$&er)ore #f A L 0 $&e" 9e s!y $&!$ P6yB def#"es ! rel!$#o" #" A or $&!$ R #s ! rel!$#o" #" A.

E-a%5#e <8

T&e ope" se"$e"%e P6yB 9&#%& re!ds 6 #s less $&!" y def#"es ! rel!$#o" #" $&e r!$#o"!l r! $#o"!l "u)bers

E-a%5#e <8

T&e ope" se"$e"%e 6 #s $&e &usb!"d of y def#"es ! rel!$#o" fro) $&e se$ of )e" $o $&e se$ of 9o)e".

3.3 3.3

>7

So#! So#!t" t"on on Se Sets ts an+ an+ G$a G$a5 5ss o0 o0 Re# Re#at at"o "ons ns

MTH 131


*e$ R L A A 0 P6yBB be ! rel!$#o". T&e Solution set '( of of $&e rel!$#o" R %o"s#s$s %o"s#s$s of $&e ele)e"$ ele)e"$ss !bB #" A 6 0 for 9&#%& P!bB P!bB #s $rue. $rue. I" o$&er 9ords Rc L N!bB X ! ∈ A b ∈ 0 P!bB #s $rue o$#%e $&!$ Rc $&e solu$#o" se$ of ! rel!$#o" R fro) A 0 #s ! subse$ of A 6 A. He"%e He"%e Rc %!" be d#spl!ye d#spl!yed d #.e plo$$e plo$$edd or s+e$%&ed s+e$%&ed o" $&e %oord#"!$e d#!'r!) of A 6 A T&e 'r!p& of ! rel!$#o" R fro) A $o 0 %o"s#s$s of $&ose po#"$s o" $&e %oord#"!$e d#!'r!) of A 6 A 9&#%& belo"'s $o $&e solu$#o" se$ of r. E-a%5#e 3<1

*e$ R L A 0 P6 yBB 9&ere A L N732 !"d 0 L N3 2 8 8 !"d P6yB P6yB re!ds re!ds 6 d#v#des d#v#des y. y. T&e" $&e solu$#o" se$ of R #s: Rc L N72B 74B 33B 34B 22B T&e solu$#o" se$ of R #s d#spl!yed o" $&e %oord#"!$e d#!'r!) of A 6 0 !s s&o9" #" #'.4.7 belo9

*e$ R be $&e rel!$#o" #" $&e re!l "u)bers def#"ed by

E-a%5#e 3<

y<6  1

>3

MTH 131


T&e T&e s&!d s&!ded ed !re! !re! #" $&e $&e %oor %oord# d#"! "!$e $e d#!' d#!'r! r!) ) of ℜ 6 ℜ s&o9" #" #'. 4.7 !bove %o"s#s$s of $&e po#"$s 9&#%& belo"' $o ℜ $&e solu$#o" se$ of R $&!$ #s $&e 'r!p& of R. o$#%e $&!$ ℜ %o"s#s$s of $&e po#"$s belo9 $&e l#"e y L 6  1. T&e l#"e l#"e y L 6 1 #s d!s&ed d!s&ed #" order order $o s&o9 s&o9 $&!$ $&!$ $&e po#"$s o" $&e l#"e do "o$ belo"' $o ℜ. 3.6 3.6

Re#a Re#at" t"on onss As As Set Setss O0 O0 O$+ O$+e$ e$e+ e+ Pa"$ Pa"$ss

*e$ Rc be !"y !"y subse$ subse$ of A 6 0. (e %!" def#" def#"ee ! rel!$#o" rel!$#o" R L A 0 P6yBB 9&ere P6yB re!d T&e ordered p!#r 6yB belo"'s $o Rc T&e solu$#o" se$ of $&#s rel!$#o" rel!$#o" R #s $&e or#'#"!l or#'#"!l se$ Rc. T&us $o every rel!$#o" R L A 0 P6yBB $&ere %orrespo"ds ! u"#ue solu$#o" se$ Rc 9&#%& #s ! subse$ of A 6 0 !"d $o every subse$ Rc of A 6 0 $&ere %orrespo"ds ! rel!$#o" R L A 0 P6yBB for 9&#%& Rc #s R L A 0 P6yBB !"d subse$s Rc of A 6 0 9e reder#"e ! rel!$#o" by $&e De0"n"t"on 8.1<

A rel!$#o" R fro) A $o 0 #s ! subse$ of A 6 0

Al$&ou'& Def#"#$#o" 4.1 of ! rel!$#o" )!y see) !r$#f#%#!l #$ &!s $&e !dv!"$!'e $&!$ 9e do "o$ use #" $&#s def#"#$#o" of ! rel!$#o" $&e u"def#"ed %o"%ep$s ope" se"$e"%e !"d v!r#!ble E-a%5#e 6.1<

*e$ A L N1 73 !"d 0 L N! b. $&e" R L N! !B 1 bB 3!B Is ! rel!$#o" rel!$#o" fro) A $o 0. ur$&er)ore 1 R ! 7 R b 3 R ! 3 R b

E-a%5#e 6.<

>2

*e$ ( L N! b %. T&e" R L N! bB !%B %%B %bB #s ! rel!$#o" #" (. (. Moreover. ! R ! b R ! % R % ! R b

MTH 131 E-a%5#e 6.3<


*e$ R L N6 yB X 6 ∈ℜ y∈ℜ y<67.

T&e" R #s ! se$ of ordered p!#rs of re!l "u)bers #.e ! subse$ of ℜ 6 ℜ. He"%e R #s ! rel!$#o" #" $&e re!l "u)bers 9&#%& %ould !lso be def#"ed by RL ℜ ℜ P6yBB (&ere P6yB re!ds y #s less $&!" 6 7 *e$ se$ A &!ve ) ele)e"$s !"d se$ 0 &!ve " ele)e"$s. T&e" $&ere !re 7)" d#ffere"$ rel!$#o"s fro) A $o 0 s#"%e A 6 0 9&#%& &!s )" ele)e"$s &!s 7 )" d#ffere"$ subse$s. Re%a$* 8.1

(e "o9 %o"s#der ! rel!$#o" be$9ee" ! se$  #"$o #$self.. Suppose SC #s ! se$ !"d le$ RCSS $&e" R #s s!#d $o ! rel!$#o" #" S. (e &!ve $&e follo9#"' proper$#es:/ 3.8

Re0#e-")e Re#at"ons

Suppose R #s ! rel!$#o" #" ! se$ S $&e" R #s refle6#ve #" S #f !"d o"ly #f for !ll 6 S 6R6 ∈

E6!)ple #B

T!+e  L IR $&e se$ of pos#$#ve re!l "u)bersB. *e$ R be $&e rel!$#o" of eu!l#$y o" IR  Is R refle6#ve o" IR W es #$ #s refle6#ve. To see $&#s le$ ! IR  $&e" ! L !. #.e. !R! ⇒ R #s refle6#ve. ∈

##B

le$ A be ! se$ !"d $!+e  L PAB or 7 A subse$ of A.

#.e. $&e %olle%$#o" of

*e$ R $o be rel!$#o" #s ! proper subse$ of  #.e. 0RD #f 0 #s ! proper subse$ of D. Is R refle6#ve o" AW >8

MTH 131


A"s9er T&e !"s9er #s "o. s#"%e ! se$ %!""o$ be ! proper subse$ of #$self #.e. A ¢A &e"%e R #s !"$#/refle6#ve. 3.4

S,%%et$"( Re#at"on

R #s sy))e$r#% #" S #f !"d o"ly #f for !ll 6y s 6Ry ⇒ yR6. ∈

I" e6!)ple #B !bove #f ! b IR  !"d ! L b $&e" b L ! #.e. !Rb bR! &e"%e R #s sy))e$r#%. ∈

⇒

I" e6!)ple ##B !bove does AR0 #)ply 0RAW o. or #f AC0 $&e" 0 ¢A hence R is not symmetric.

3.>

Ant"=S,%%et$"( Re#at"on

R #s !"$#/sy))e$r#% #" S #ff for 6y S 6Ry !"d yR6 ∈

I" e6!)ple #B !bove !Rb !"d bR! 3.?

⇒

⇒

6 L y.

!Lb &e"%e R #s !"$#/sy))e$r#%.

T$ans"t")e Re#at"on

R #s ! $r!"s#$#ve rel!$#o" o" s #ff for !ll 6y S 6Ry yR ⇒ 6P. ∈

Co"s#der e6!)ple ##B !bove le$ AR0 !"d 0RD does ARD &oldsW es for #f AC0 !"d 0CD $&e" ACD $&erefore R #s $r!"s#$#ve. 3.=

E>!")a#en(e Re#at"on

A rel!$#o" E #" ! se$ S #s s!#d $o be !" eu#v!le"%e rel!$#o" #" S #f for !ll 6y s ∈

#B ##B ###B

6E6 refle6#ve proper$yB 6Ey yE ⇒ yE6 sy))e$r#%B 6Ey yE → 6E $r!"s#$#veB

or e6!)ple $&e rel!$#o" of eu!l#$y def#"ed #" e6!)ple #B #s !" eu#v!le"%e rel!$#o" %&e%+B. >4

MTH 131

3.1<


Do%a"n an+ Range o0 a Re#at"on

*e$ R be ! rel!$#o" be$9ee"  !"d . T&e" R  #s ! rel!$#o" be$9ee"  !"d . T&us 6 R /1y #ffy R6. (e $&erefore def#"e #B ##B ###B

R!"'e of R L Do)!#" R /1B R!"'e of R/1 L Do)!#" RB ur$&er)ore R /1B/1 L R

3.11

Re#at"ons an+ F!n(t"on

T&e "o$#o" of rel!$#o" #s ! 'e"er!l#!$#o" of $&e "o$#o" of fu"%$#o"s. *e$ f:  →  be fu"%$#o" def#"ed o"  !"d . $&e" f #s ! fu"%$#o" #f #$ %!" s!$#sfy $&e follo9#"' s#"'le v!lued proper$y: 6yB∈ f 6B ∈ f 6.

⇒

y L 

CONCLUSION

I" $&#s u"#$ e)p&!s#s &!s bee" pl!%e o" $&e der#v!$#o" of rel!$#o"s !s ordered p!#rs be$9ee" $9o se$s b!sed o" ope" se"$e"%esF f#"d#"' $&e Do)!#" R!"'e !"d I"verse of ! rel!$#o"F def#"#"' 9#$& e6!)ples $&e d#ffere"$ +#"ds of rel!$#o"s !"d s$!$#"' 9&e$&er or "o$ ! rel!$#o" def#"ed o" ! se$ #s ! fu"%$#o" of $&e se$ #"$o #$self. Here #s $&e su))!ry 7.

SUMMARY

• T&e e6press#o" P6yB #s %!lled !" open sentence in two variables x and y.

• R L A 0 P!bBB #s %!lled relation fro) A $o 0 9&ere !bB

∈ A 6 0

• Rc L N!bB X !∈A b∈0 P!bB #s $rue #s $&e solution set of $&e

rel!$#o" R. • A rel!$#o" R fro) A $o 0 #s ! subse$ of A 6 0

>>

MTH 131


• R 1 L Nb!B X !bB ∈ R#s $&e inverse relation of R • If every ele)e"$ #" ! se$ #s rel!$ed $o #$self $&e" $&e rel!$#o" #s s!#d • • • • • •

$o be refle)ive or ! rel!$#o" R #" A #f !bB ∈ R #)pl#es b!B ∈ R $&e" R #s ! symmetric rel!$#o" or ! rel!$#o" R #" A #f !bB ∈ R !"d b!B ∈ R #)pl#es ! L b $&e" R #s !" anti-symmetric relation or ! rel!$#o" R #" A #f !bB ∈ R !"d b%B ∈ R #)pl#es !%B ∈ R $&e" R #s ! transitive relation A rel!$#o" #" ! se$ #s !" equivalence relation #f #$ #s refle6#ve $r!"s#$#ve !"d sy))e$r#% D L N! X ! ∈ A !bB ∈ R $&e se$ of !ll f#rs$ ele)e"$s of $&e ordered p!#rs 9&#%& belo"' $o $&e rel!$#o" fro) A $o 0 #s $&e domain of $&e rel!$#o". E L NbXb∈0 !bB ∈ R $&e se$ of !ll se%o"d ele)e"$s of $&e ordered p!#rs 9&#%& belo"'s $o $&e rel!$#o" fro) A $o 0 #s $&e range of $&e rel!$#o"

8.

TUTOR=MAR?ED ASSIGNMENT

1.

Co"s#der $&e rel!$#o" R L N18B 28B 12B 24B 3>B >4B #"d 1B $&e do)!#" R 7B $&e r!"'e of R 3B $&e #"verse R.

7.

*e EL N17 3. Co"s#der $&e follo9#"' rel!$#o" #" E.

9.

REFERENCE AND FURTHER READINGS

Sey)our *#ps%&u$F S%&!u)Js Ou$l#"e Ser#es: T&eory !"d Proble)s of Se$ T&eory !"d rel!$ed $op#%s 1=42pp. 1  133. Su"d!y O. Iy!&e"F I"$rodu%$#o" $o Re!l A"!lys#s Re!l/v!lued fu"%$#o"s of ! re!l v!r#!ble 1==? 5ol. 1B

>?

MTH 131

UNIT 


FURTHER THEORY OF SETS

CONTENTS

1.< 7.< 3.<

2.< 8.< 4.< >.<

I"$rodu%$#o" Ob,e%$#ves M!#" 0ody 3.1 Al'ebr! of Se$s 3.7 I"de6ed Se$s 3.3 Ge"er!l#ed Oper!$#o"s 3.2 P!r$#$#o"s 3.8 Eu#v!le"%e Rel!$#o"s !"d P!r$#$#o"s Co"%lus#o" Su))!ry Tu$or M!r+ed Ass#'")e"$s Refere"%es !"d ur$&er Re!d#"'s

1.

INTRODUCTION

I" $&#s u"#$ 9e l!y ! b!s#% fou"d!$#o" of ! br!"%& of )!$&e)!$#%s *o'#%B $&!$ s$ud#es l!9s !sso%#!$ed 9#$& $&e se$ oper!$#o"sF #"$erse%$#o" u"#o" !"d %o)ple)e"$. ou 9#ll do 9ell $o follo9 %losely $&e re!so"#"' prese"$ed #" $&e $e6$. .

O4ECTIVES

Af$er 'o#"' $&rou'& $&#s u"#$ you s&ould be !ble $o do $&e follo9#"':     

3.

Prove #de"$#$#es us#"' $&e $!ble of l!9s of $&e !l'ebr! of se$s #"d $&e du!l of !"y #de"$#$y Ge"er!l#e $&e oper!$#o"s of u"#o" !"d #"$erse%$#o"s of se$s #"d $&e poss#ble p!r$#$#o"s of ! se$ S$!$e $&e rel!$#o"s&#p be$9ee" Eu#v!le"%e rel!$#o"s !"d P!r$#$#o"s

MAIN ODY

>=

MTH 131

3.1


A#ge&$a o0 Sets

Se$ u"der $&e oper!$#o"s of u"#o" #"$erse%$#o" !"d %o)ple)e"$ s!$#sfy v!r#ous l!9s #.e. #de"$#$#es. 0elo9 #s ! $!ble l#s$#"' l!9s of se$s )os$ of 9&#%& &!ve !lre!dy bee" "o$ed !"d prove" #" u"#$ 7. O"e br!"%& of )!$&e)!$#%s #"ves$#'!$e $&e $&eory of se$ by s$udy#"' $&ose $&eore)s $&!$ follo9 fro) $&ese l!9s #.e. $&ose $&eore)s 9&ose proofs reu#re $&e use of o"ly $&ese l!9s !"d "o o$&ers. (e 9#ll refer $o $&e l!9s #" T!ble 1 !"d $&e#r %o"seue"%es !s $&e !l'ebr! of se$s. LAWS OF THE ALGERA OF SETS I+e%5otent La/s

1!.

A ∪ A L A

1b.

A ∩ A L A

Asso("at")e La/s

7!.

A ∪ 0B∪C L A∪0∪CB

7b.

A∩0B∩CL A∩0∩CB

Co%%!tat")e La/s

3!.

A∪0 L 0∪A

3b.

A∩0 L 0∩A

D"st$"&!t")e La/s

2!.

A∪0∩CBLA∪0B∩A∪CB 2b.

A∩0∪CBLA∩0B∪A∩CB

I+ent"t, La/s

8!. 4!.

A∪∅ L A A∪U L U

8b. 4b.

A∩U L ∅ A∩∅ L ∅

Co%5#e%ent La/s

>!. ?!.

A∪AJ L U AJBJ L A

>b. ?b.

A∩AJ L ∅ UJ L ∅ ∅J L U

De Mo$gan@s La/s

=!.

A∪0BJ L AJ∩0J

=b. Ta&#e 1

?<

A∩0BJ L AJ∪0J

MTH 131


o$#%e $&!$ $&e %o"%ep$ of ele)e"$ !"d $&e rel!$#o"  ! belo"'s $o A do "o$ !ppe!r !"y9&ere #" $!ble 1. Al$&ou'& $&ese %o"%ep$s 9&ere esse"$#!l $o our or#'#"!l develop)e"$ of $&e $&eory of se$s $&ey do "o$ !ppe!r #" #"ves$#'!$#"' $&e !l'ebr! of se$s. T&e rel!$#o" A #s ! subse$ of 0 #s def#"ed #" our !l'ebr! of se$s by. A ⊂ 0 )e!"s A ∩ 0 L A As e6!)ples 9e "o9 prove $9o $&eore)s #" our !l'ebr! of se$s $&!$ #s 9e prove $9o $&eore)s 9&#%& follo9 d#re%$ly fro) $&e l!9s #" T!ble 1. O$&er $&eore)s !"d proofs !re '#ve" #" $&e proble) se%$#o". E-a%5#e 1.1 Prove: A ∪0B ∩ A ∪ 0JB L A Statement

1. 7. 3. 2. 8.

A∪0B ∩ A∪0JB L A∪0∩0JB 0 ∩ 0J L ∅ ∴ A∪0B∩A∪0JB L A∪∅ A ∪∅ L A ∴ A∪0B∩A∪0JB L A

E-a%5#e 1.<

8.

1.D#s$r#bu$#ve *!9 7. Subs$#$u$#o" 3. Asso%#!$#ve *!9 2. Subs$#$u$#o" 8. Def#"#$#o" of subse$

Prove A ⊂ 0 !"d 0 ⊂ C #)pl#es A ⊂ C.

Statement

1. 7. 3. 2.

'eason

A L A ∩ 0 !"d 0 L 0 ∩ C ∴ A L A ∩ 0 ∩ CB A L A ∩ 0B ∩ C ∴ A L A ∩ C ∴ A ⊂ C

'eason

1. 7. 3. 2.

Def#"#$#o" of subse$ Subs$#$u$#o" Asso%#!$#ve *!9 Subs$#$u$#o"

8. Def#"#$#o" of subse$

P$"n("5#e o0 D!a#"t,

If 9e #"$er%&!"'e ∩ !"d ∪ !"d ∅ #" !"y s$!$e)e"$ !bou$ se$s $&e" $&e "e9 s$!$e)e"$ #s %!lled $&e du!l of $&e or#'#"!l o"e. E-a%5#e .1<

T&e du!l of U ∪ 0B ∩ A ∪ ∅B L A ?1

MTH 131


#s

∅ ∩ 0B∪A ∩ UB L A

o$#%e $&!$ $&e du!l of every l!9 #" T!ble 1 #s !lso ! l!9 #" T!ble 1. T&#s f!%$ #s e6$re)ely #)por$!"$ #" v#e9 of $&e follo9#"' pr#"%#ple: P$"n("5#e o0 D!a#"t,<

If %er$!#" !6#o)s #)ply $&e#r o9" du!ls $&e" $&e du!l of !"y $&eore) $&!$ #s ! %o"seue"%e of $&e !6#o)s #s !lso ! %o"seue"%e of $&e !6#o). or '#ve" !"y $&eore) !"d #$s proof $&e du!l of $&e $&eore) %!" be prove" #" $&e s!)e 9!y by us#"' $&e du!l of e!%& s$ep #" $&e or#'#"!l proof.

T&us $&e pr#"%#ple of du!l#$y !ppl#es $o $&e !l'ebr! of se$s E-a%5#e .<

Prove: A ∩ 0B ∪ A ∩ 0JB L A

T&e du!l of $&#s $&eore) #s prove" #" E6!)ple 1.1F &e"%e $&#s $&eore) #s $rue by $&e Pr#"%#ple of Du!l#$y. 3.

In+e-e+ Sets

Co"s#der $&e se$s A1 L N1 1< A 7 L N7 2 4 1< A 3 L 3 4 = A2 L N2 ? A 8 L N8 4 1< A"d $&e se$

I L N1 7 3 2 8

o$#%e $&!$ $o e!%& ele)e"$ # ∈ I $&ere %orrespo"ds ! se$ A#. I" su%& ! s#$u!$#o" I #s %!lled $&e inde) set, $&e se$s A1@.. A8B !re %!lled $&e inde)ed sets, !"d $&e subs%r#p$ # of A # #.e. e!%& # ∈ I  #s %!lled !" inde) . ur$&er)ore su%& !" #"de6ed f!)#ly of se$s #s de"o$ed by A#B#∈I (e %!" loo+ !$ !" #"de6ed f!)#ly of se$s fro) !"o$&er po#"$ of v#e9. S#"%e $o e!%& ele)e"$ # ∈ I $&ere #s !ss#'"ed ! se$ A # 9e s$!$e

?7

MTH 131 De0"n"t"on 9.1<


A" #"de6ed f!)#ly of se$s A #B # ∈ I #s ! fu"%$#o" f: I → A (&ere $&e do)!#" of f #s $&e #"de6 se$ I !"d $&e r!"'e of f #s ! f!)#ly of se$s.

E-a%5#e 3.1<

Def#"e 0" L N6 < ≤ 6 ≤ 1"B 9&ere " ∈  $&e "!$ur!l "u)bers. T&e" 01 L _<1` 07 L _<17` . . .

E-a%5#e 3.<

*e$ be $&e se$ of 9ords #" $&e E"'l#s& *!"'u!'e !"d le$ # ∈ I . Def#"e (1 L N6 6 #s ! le$$er #" $&e 9ord # ∈ I . If # #s $&e 9ord follo9  $&e" ( # L Nf 1 o 9 .

E-a%5#e 3.3<

Def#"e D" L N6 6 #s ! )ul$#ple of " 9&ere " ∈  $&e "!$ur!l "u)bers. T&e" D1 L N1 7 3 2 . . .  D 7 L N7 2 4 ? . . .  D 3 L N3 4 = 17 . . .  o$#%e $&!$ $&e #"de6 se$ N #s !lso D1 !"d !lso $&e u"#vers!l se$ fro $&e #"de6ed se$s.

Re%a$* 9.1<

A"y f!)#ly of 0 of se$s %!" be #"de6ed by #$self. Spe%#f#%!lly $&e #de"$#fy fu"%$#o" #: 0 → 0 #s !" #"de6ed f!)#ly of se$s NA#  1 ∈ 0 (&ere A # ∈ 0 !"d 9&ere # L A #. I" o$&er 9ords $&e #"de6ed of !"y se$ #" 0 #s $&e se$ #$self

?3

MTH 131

3.3


Gene$a#"e+ O5e$at"ons

T&e oper!$#o" of u"#o" !"d #"$erse%$#o" 9ere def#"ed for $9o se$s. T&ese def#"#$#o"s %!" e!s#ly be e6$e"ded by #"du%$#o" $o ! f#"#$e "u)ber of se$s. Spe%#f#%!lly for se$s A 1 . . . A " u# L I A1 ≡ A1 ∪ A7 ∪ K ∪ A" I" # L I Ai ≡ A1 ∩ A7 ∩ K ∩ A" I" v#e9 of $&e !sso%#!$#ve l!9 $&e u"#o" #"$erse%$#o"B of $&e se$s )!y be $!+e" #" !"y orderF $&us p!re"$&eses "eed "o$ be used #" $&e !bove. T&ese %o"%ep$s !re 'e"er!l#sed #" $&e follo9#"' 9!y. Co"s#der $&e #"de6ed f!)#ly of se$s NA## ∈ I ∪ # ∈ J A# A"d le$ J I. T&e" ⊂

Co"s#s$s of $&ose ele)e"$s 9&#%& belo"' $o !$ le!s$ o"e A # 9&ere # ∈ J. Spe%#f#%!lly ∪ # ∈ J A# L N 6  $&ere e6#s$s !" # ∈ J . su%& $&!$ 6 ∈ A# ∪ # ∈ J A#

I" !" !"!lo'ous 9!y

%o"s#s$ of $&ose ele)e"$s 9&#%& belo"' $o every A # for # ∈ J . I" o$&er 9ords ∩ # ∈ J A# L N6 6 ∈ A# for every # ∈ J  E-a%5#e 6.1<

*e$ A1 L N1 1< A 7 L N7 2 4 1< A 3 L N3 4 =  A2 L N2 ? A8 L N8 4 1<F !"d le$ J L N7 3 8 . T&e" ∩ # ∈ J A# L N 4 !"d ∪ # ∈ J A# L N7 2 4 1< 3 = 8

E-a%5#e 6.<

*e$ 0" L _< #"` 9&ere " ∈ N  $&e "!$ur!l "u)bers. T&e" ∩ # ∈ N 0# L N< !"d ∪ # ∈ N 0# L _< 1`

E-a%5#e 6.3<

*e$ D" L N6  6 #s ! )ul$#ple of " 9&ere " ∈  $&e "!$ur!l "u)bers. T&e"

?2

∩ #∈  D1 L ∅

MTH 131


T&ere !re !lso 'e"er!l#sed d#s$r#bu$#ve l!9s for ! se$ 0 !"d !" #"de6ed f!)#ly of se$s NA ## ∈ I be !" #"de6ed f!)#ly of se$s. T&e" for !"y se$ 0 0 ∩  ∪ #∈I A#B L ∪ #∈I 0 ∩ A#B 0 ∪ ∩#∈I A#B L ∩ #∈I 0 ∪ A#B 3.6

Pa$t"t"ons

Co"s#der $&e se$ A L N 1 7 . . . = 1< !"d #$s subse$s 01 L N1 3 07 L N> ? 1< 0 3 L N7 8 4.02 L 2 = T&e f!)#ly of se$s 0 L N0 1 07 03 02 &!s $9o #)por$!"$ proper$#es. 1. 7.

A #s $&e u"#o" of $&e se$s #" 0 #.e. A L 01 ∪ 07 ∪ 03 ∪ 02 or !"y se$s 0# !"d 0 , E#$&er 0# L 0 , or 0# ∩ 0 ,L ∅

Su%& ! f!)#ly of se$s #s %!lled partition of A. Spe%#f#%!lly 9e s!y De0"n"t"on 9.<

*e$ N0# #∈I be ! f!)#ly of "o"/e)p$y subse$s of A. T&e" N0 # #∈I #s %!lled ! partition of A #f P1: ∪ #∈I 0# L A P7: or !"y 0# 0 , e#$&er 0 , or 0# ∩ 0 , L ∅ ur$&er)ore e!%& 0# #s $&e" %!lled !" equivalence class of A. E-a%5#e 7.1<

*e$  L N1 7 3 . . . E L N7 2 4 . . . !"d  L N1 3 8. T&e" Ne  #s ! p!r$#$#o" of .

E-a%5#e 7.<

*e$ T L N1 7 3. . . = 1< !"d le$ A L N1 3 8 0 L N7 4 1< !"d C L N2 ? =. T&e" NA 0 C #s "o$ ! p!r$#$#o" of T s#"%e T ≠ A ∪ 0 ∪ C #.e. s#"%e > ∈ T bu$ > ≠  A ∪ 0 ∪ CB. ?8

MTH 131


*e$ T L N1 7 @ = 1< !"d  L N1 3 8 > = !"d G L N7 2 1< !"d H L N3 8 4 ?. T&e" Nf G H #s "o$ ! p!r$#$#o" of T s#"%e

E-a%5#e 7.3<

 ∩ H ≠ ∅  ≠ H le$ y1 y7 y3 !"d y2 be respe%$#vely $&e 9ords follo9 $&u)b flo9 !"d !'!#" !"d le$

E-a%5#e 7.6<

A L N9 ' u o I ) ! $ f " & b ur$&er)ore def#"e (# L N6 6 #s $&e le$$er #" $&e 9ord y # T&e" N(# (7 (3 (2 #s $&e p!r$#$#o" of A. "o$#%e $&!$ (1 !"d (3 !re "o$ d#s,o#"$ bu$ $&ere #s "o %o"$r!d#%$#o"s s#"%e $&e se$s !re eu!l. 3.7

E>!")a#en(e Re#at"ons an+ Pa$t"t"on

Re%!ll $&e follo9#"' De0"n"t"on<

A rel!$#o" #" ! se$ A #s !" eu#v!le"$ rel!$#o" #f < 1. R #s refle6#ve #.e. for every ! ∈ A ! #s rel!$ed $o #$selfF 7. R #s sy))e$r#% #.e. #f ! #s rel!$ed $o b $&e" b #s rel!$ed $o !F 3. R #s $r!"s#$#ve #.e. #f ! #s rel!$ed $o be !"d b #s rel!$ed $o % $&e" ! #s rel!$ed $o %.

T&e re!so" $&!$ p!r$#$#o" !"d eu#v!le"%e rel!$#o"s !ppe!r $o'e$&er #s be%!use of $&e Teo$e% 9.< F!n+a%enta# Teo$e% o0 E>!")a#en(e Re#at"ons< *e$

R be !" eu#v!le"%e rel!$#o" #" ! se$ A !"d for every ! ∈ A le$ 0∞ L N6 6 αB ∈ RB

?4

MTH 131


#.e. $&e se$ of ele)e"$s rel!$ed $o α. T&e" $&e f!)#ly of se$s N0α  α ∈ A #s ! p!r$#$#o" of A. I" order 9ords !" eu#v!le"%e rel!$#o" R #" ! se$ A p!r$#$#o" $&e se$ A by pu$$#"' $&ose ele)e"$s 9&#%& !re rel!$ed $o e!%& #" $&e s!)e eu#v!le"%e %l!ss. Moreover $&e se$ 0α #s %!lled equivalence class de$er)#"ed by α !"d $&e se$ of eu#v!le"%e %l!sses N0 α∈αA #s de"o$ed by A R A"d %!lled quotient set . T&e %o"verse of $&e prev#ous $&eore) #s !lso $rue. Spe%#f#%!lly Teo$e% 9.3:

*e$ N0# #∈I be ! p!r$#$#o" of A !"d le$ R be $&e rel!$#o" #" A def#"ed by $&e ope" se"$e"%e 6 #s #" $&e s!)e se$ of $&e f!)#ly N0 # #∈I !s y. T&e" R #s !" eu#v!le"%e rel!$#o" #" A.

T&us $&ere #s ! o"e $o o"e %orrespo"de"%e be$9ee" !ll p!r$#$#o"s of ! se$ A !"d !ll eu#v!le"%e rel!$#o"s #" A. E-a%5#e 8.1<

I" $&e Eu%l#de!" pl!"e s#)#l!r#$#es of $r#!"'les #s !" eu#v!le"%e rel!$#o". T&us !ll $r#!"'les #" $&e pl!"e !re por$#o"ed #"$o d#s,o#"$ se$s #" 9&#%& s#)#l!r $r#!"'les !re ele)e"$s of $&e s!)e se$.

E-a%5#e 8.<

*e$ R 8 be $&e rel!$#o" #" $&e #"$e'ers def#"ed by - , )od 8B

9&#%& re!ds 6 #s %o"'rue"$ $o y )odulo 8 !"d 9&#%& )e!"s 6 y #s d#v#s#ble eu#v!le"%e %l!sses #" [ R 8: E< E1 E7 E3 !"d E2. S#"%e e!%& #"$e'er 6 #s u"#uely e6press#ble #" $&e for) 6 L 8p  r 9&ere < ≤ r ] 8 $&e" 6 #s ! )e)ber of $&e eu#v!le"%e %l!ss Er 9&ere r #s $&e re)!#"der. ?>

MTH 131

T&us


E< L N. . . /1< /8 < 8 1< . . . E1 L N. . .  /= /2 1 4 11 . . . E7 L N. . .  /? /3 7 > 17 . . .  E3 L N. . .  /> /7 3 ? 13 . .  E3 L N. . .  /4 1/  2 = 12 . . . Add $&e uo$#e"$ se$ [R 8 L NE< E1 E7 E3 E2

6.

CONCLUSION

ou !re 'r!du!lly be#"' #"$rodu%ed $o us#"' se$ "o$!$#o" r!$&er $&!" s$!$e)e"$s. T&eore)s #" $&e !l'ebr! of se$s !re )os$ useful #" prov#"' #de"$#$#es rel!$ed $o lo'#% !"d re!so"#"' #" )os$ %!ses us#"' $&e pr#"%#ple of du!l#$y. 7.

SUMMARY

I" #"ves$#'!$#"' $&e !l'ebr! of se$s you "eed $o $!+e "o$e of $&e du!l of $&e s$!$e)e"$s #" $!ble 1. A" #"de6ed f!)#ly of se$s NA # #∈I #s su%& $&!$ for e!%& #"de6 # L 1 7 3 2 . . . 9e &!ve se$s A 1 A7 A3 . . . *e$ N0# #∈I be ! f!)#ly of "o"/ e)p$y subse$s of A. $&e" N0 # #∈I #s %!lled partition of A #f ∪ #∈I 0# L A !"d for !"y 0 # 0 , e#$&er 0# L 0 , or 0# ∩ 0 , L∅ ur$&er)ore e!%& 0# #s $&e" %!lled !" equivalence class of A.

8.

TUTOR MAR?ED ASSIGNMENTS

1.

(r#$e $&e du!l of e!%& of $&e follo9#"': 1. 0 ∪ CB ∩ AL 0 ∩ AB ∪ C ∩ AB 7. A ∪ AJ ∩ 0BL A ∪ 0

??

MTH 131

3.


A ∩ UB ∩  ∅ ∪ AJB L ∅

7

Prove: A ∩ 0B ∪ A ∩ 0JB L A.

3.

*e$ 0# L _# #  1` 9&ere I ε [ $&e #"$e'er. f#"d 1. 7. 3. 2.

2.

*e$ A L N! b % d e f '. S$!$e 9&e$&er or "o$ e!%& of $&e follo9#"' f!)#l#es of se$s #s ! p!r$#$#o" of A. 1. 7. 3. 2.

9.

01 ∪ 07 03 ∪ 02 ∪1?# L > Bi ∪# ε [ 0#

N01 L N! % e 0 7 L Nb 03 L Nd ' NC1 L N! e ' C 7 L N% d C3 L Nb e f ND1 L N! b e ' D 7 L N% D3 L Nd f NE1L N! b % d e f '


Sey)our *#ps%&u$F S%&!u)Js Ou$l#"e Ser#es: T&eory !"d Proble)s of Se$ T&eory !"d rel!$ed $op#%s 1=42. pp. 1  133. Su"d!y O. Iy!&e"F I"$rodu%$#o" $o Re!l A"!lys#s Re!l/v!lued fu"%$#o"s of ! re!l v!r#!ble 1??= 5ol. 1

UNIT 3

FURTHER THEIRY OF FUNCTIONSB OPERATIONS

CONTENTS

?=

MTH 131


1. C&!r!%$er#s$#% u"%$#o"s 3.?. Oper!$#o"s 3.?.1 Co))u$!$#ve Oper!$#o"s 3.?.7 Asso%#!$#ve Oper!$#o"s 3.?.3 D#s$r#bu$#ve Oper!$#o"s 3.?.2 Ide"$#$y Ele)e"$s 3.?.8 I"verse Ele)e"$s 3.= Oper!$#o"s !"d Subse$s 2.< Co"%lus#o" 8.< Su))!ry 4.< Tu$or/M!r+ed Ass#'")e"$s >.< Refere"%es !"d ur$&er Re!d#"'s 1.

INTRODUCTION

T&ere !re so)e fur$&er %o"%ep$s you &!ve $o be%o)e f!)#l#!r 9#$& "o9 9&#%& you 9#ll %o)e !%ross #" )!$&e)!$#%!l !"!lys#s !"d !bs$r!%$ )!$&e)!$#%s. T&#s u"#$ #"$rodu%es you $o so)e of $&e). P!y !$$e"$#o" "o$ o"ly $o $&e def#"#$#o"s bu$ !lso $o $&e e6!)ples '#ve".

.

O4ECTIVES

A$ $&e e"d of $&#s u"#$ you s&ould be !ble $o: / s$!$e 9&e$&er or "o$ ! d#!'r!) of fu"%$#o"s #s %u)ul!$#ve fro) $&e !rro9s %o""e%$#"' $&e fu"%$#o"s. / E6pl!#" $&e $er)s Res$r#%$#o" !"d E6$e"s#o" of fu"%$#o"s =<

MTH 131


/ Des%r#be $&e follo9#"': Se$ fu"%$#o"s Re!l/v!lued fu"%$#o"s !"d #$s !l'ebr! C&!r!%$er#s$#% fu"%$#o" / Apply $&e Rule of $&e M!6#)u) Do)!#". / E6pl!#" oper!$#o"s o" C!r$es#!" produ%$s 3.

MAIN ODY

3.1

F!n(t"ons an+ D"ag$a%s

As )e"$#o"ed prev#ously $&e sy)bol A 0 De"o$es ! fu"%$#o" of A #"$o 0. I" ! s#)#l!r )!""er $&e d#!'r!) &

C

f

' 0

Co"s#s$s of le$$ers A 0 !"d C de"o$#"' se$s !rro9s f ' & de"o$#"' fu"%$#o"s f: A → 0 ': 0 → C !"d &: A → C !"d $&e seue"%e of !rro9s Nf ' de"o$#"' $&e %o)pos#$e fu"%$#o" ' o f: A → C. E!%& of $&e fu"%$#o"s &: A →C !"d ' o f: A → C $&!$ #s e!%& !rro9 or seue"%e of !rro9s %o""e%$#"' A $o C #s %!lled ! p!$& fro) A $o C. De0"n"t"on :. 1<

A d#!'r!) of fu"%$#o"s #s s!#d $o be %u)ul!$#ve #f for !"y se$  !"d  #" $&e d#!'r!) !"y $9o p!$&s fro)  $o  !re eu!l.

E-a%5#e 1.1<

Suppose $&e follo9#"' d#!'r!) of fu"%$#o"s #s Cu)ul!$#ve. D =1

MTH 131


#

& A

C f

0

'

T&e" # o & L f ' o I L , !"d ' o f L , o & L ' o I o &. T&e fu"%$#o"s f: A 0 !"d ': 0 A !re #"verses #f !"d o"ly #f $&e follo9#"' d#!'r!)s !re %u)ul!$#ve:

E-a%5#e 1.<

1A A

A 1A

1R

0

f

'

f

0

' 0

A

Here 1A !"d 10 !re $&e #de"$#$y fu"%$#o"s. 3.

Rest$"(t"ons An+ E-tens"ons O0 F!n(t"ons

*e$ f be ! fu"%$#o" of A #"$o C #.e. le$ f: A→ C !"d le$ 0 be ! subse$ of A. T&e" f #"du%es ! fu"%$#o" fJ: 0 → C 9&#%& #s def#"ed by bB L fbB or !"y b∈0. $&e fu"%$#o" fJ #s %!lled $&e res$r#%$#o" of  $o 0 !"d #s de"o$ed by  0

E-a%5#e .1<

=7

*e$ f: ℜ → ℜ be def#"ed by f6B L 6 7. T&e"

MTH 131


  L N1 1B 72B 3=B 2 14B @ I#s $&e res$r#%$#o" of $o  $&e "!$ur!l "u)bers. E-a%5#e .<

T&e se$ ' L N7 8B 8 1B 3 >B ? 3B = 8B #s ! fu"%$#o" fro) N7 8 3 ? = #"$o . T&e" N7 8B 3 >B = 8B

! subse$ of ' #s $&e res$r#%$#o" of ' $o N7 3 = $&e se$ of f#rs$ ele)e"$s of $&e ordered p!#rs #" '. (e %!" loo+ !$ $&#s s#$u!$#o" fro) !"o$&er po#"$ of v#e9. *e$ f: A → C !"d le$ 0 be ! superse$ of !. T&e" ! fu"%$#o" : 0 → C #s %!lled !" e-tens"on of f #f for every ! ∈ A !B L f!B *e$ f be $&e fu"%$#o" o" $&e pos#$#ve re!l "u)ber def#"ed by f6B L 6 $&!$ #s le$ $&e #de"$#$y fu"%$#o". T&e" $&e !bsolu$e v!lue fu"%$#o" E-a%5#e .3<

 L

 #f 6 ≥ < / 6 #f 6 < <

Is !" e6$e"s#o" of f $o !ll re!l "u)bers.

E-a%5#e .6<

Co"s#der $&e fu"%$#o"

 L N1 7B 3 2B >B (&ose do)!#" #s N1 3 >. T&e" $&e fu"%$#o"  L N1 >B 3 2B 8 4B > 7B =3

MTH 131


(&#%& #s ! superse$ of $&e fu"%$#o" f #s !" e6$e"s#o" of f. 3.3

Set F!n(t"ons

*e$ f be ! fu"%$#o" of A #"$o 0 !"d le$ T be ! subse$ of A $&!$ #s A$ 0 !"d T ⊂ A. T&e"

TB

(&#%& #s re!d f of T #s def#"ed $o be $&e se$ of #)!'e po#"$ of ele)e"$s #" T. I" o$&er 9ords TB L N6 f!B L 6 ! ∈ T 6 ∈ 0 o$#%e $&!$ fTB #s ! subse$ of 0. E-a%5#e 3.1<

*e$ A L N! b % d T L Nb % !"d 0 L N1 7 3. Def#"e f: A

0 by

A 0

1

C

7

D

3

T&e" f TB L N7 3. E-a%5#e 3.<

*e$ ': ℜ → ℜ be def#"ed by '6B L 6 7 !"d le$ T L _3 2`. T&e" GTB L _= 14` L N6 = ≤ 6 ≤ 14.

=2

MTH 131


o9 le$ A be $&e f!)#ly of subse$s of A !"d le$ 0 be $&e f!)#ly of subse$s of 0 #f f: A → 0 $&e" f !ss#'"s $o e!%& se$ T ∈ A ! u"#ue se$ of fTB ∈ 0. I" o$&er 9ords $&e fu"%$#o" f: A → 0 #"du%es ! fu"%$#o" f: A → 0. Al$&ou'& $&e s!)e le$$er de"o$es e!%& fu"%$#o" $&ey !re esse"$#!lly $9o d#ffere"$ fu"%$#o"s. o$#%e $&!$ $&e do)!#" of f: A → 0 %o"s#s$s of se$s. Ge"er!lly spe!+#"' ! fu"%$#o" #s %!lled ! set %o"s#s$s of se$s. 3.6

0!n(t"on

#f #$s do)!#"

Rea#=Va#!e+ F!n(t"ons

A fu"%$#o" f: A→ ℜ 9&#%& )!ps ! se$ #"$o $&e re!l "u)bers #.e. 9&#%& !ss#'"s $o e!%& ! ∈ A ! re!l "u)ber f!B ∈ ℜ #s a $ea#=)a#!e+ 0!n(t"on. T&ose fu"%$#o"s 9&#%& !re usu!lly s$ud#ed #" ele)e"$!ry )!$&e)!$#%s e.'. P6B L !o6"  !16"/16  !" $6B L s#" 6 %os 6 or $!" 6 f6B L lo' 6 or e 6 T&!$ #s poly"o)#!ls $r#'o"o)e$r#% fu"%$#o"s !"d lo'!r#$&)#% !"d e6po"e"$#!l fu"%$#o"s !re spe%#!l e6!)ples of re!l/v!lued fu"%$#o"s. 3.7

A#ge$&$a An+ Rea#=Va#!e+ F!n(t"ons

*e$  D be $&e f!)#ly of !ll re!l/v!lued fu"%$#o"s 9#$& $&e s!)e do)!#" D. T&e" )!"y !l'ebr!#%B oper!$#o"s !re def#"ed #"  D. Spe%#f#%!lly le$ f: D ℜ !"d ': D → ℜ !"d le$ + ∈ ℜ. T&e" e!%& of $&e follo9#"' fu"%$#o"s #s def#"ed !s follo9s: f  +B:

D

ℜ

by

f  +B6B L f6B  +

  f B :

D

ℜ

by

 f  B6B L  f6B

fB:

D

ℜ

by

 fB6B L

f ± 'B:

D

ℜ

by

f ± 'B6B Lf6B ± '6B

+fB:

D

ℜ

by

+fB6B L +f6BB

f6BB

=8

MTH 131


f'B:

D

ℜ

by

f'B6B L f6B'6B

f'B:

D

ℜ

by

f'B6B Lf6B'6B

9&ere '6B ≠ < ℜ #s "o$ $&e s!)e !s $&e %o)pos#$#o" fu"%$#o" o$e $&!$ f'B: D 9&#%& 9!s d#s%ussed prev#ously. E-a%5#e 6.1<

ℜ !"d ': D

*e$ D L N! b !"d le$ f: D def#"ed by :

ℜ be

!B L 1 fbB L 3 !"d '!B L 7 'bB L /1 I" o$&er 9ords  L N! 1B b 3B !"d ' L N! 7B b /1B T&e" 3f  7'B!B L 3f!B  7'!B L 31B /77B L/1 3f  7'BbB L 3fbB  7'bB 33B  ' /1B L 11 $&!$ #s

3f  7' L N! 1B b 11B

Also s#"%e ' 6B L '6B !"d '  3B6B L '6B  3 G LN! 7B b 1B AD G  3 L NNA 8B b 7B

E-a%5#e 6.<

*e$ f: ℜ ℜ !"d ': ℜ 0y $&e for)ul!s

ℜ be def#"ed

6B L 76 L / 1 !"d '6B L 7 T&e for)ul!s 9&#%& def#"e $&e fu"%$#o" 3f  7'B: ℜ ℜ !"d f'B: ℜ ℜ !re fou"d !s follo9s: =4

MTH 131


3f  7'B6B L 376  1B /76 7B L / 767  46  3 f'B6B L 76  1B6 7B L 767 / 67 3.8

R!#e O0 Te Ma-"%!% Do%a"n

A for)ul! of $&e for) 6B L 16 '6B L s#" 6 &6B L √6 Does "o$ #" #$self def#"e ! fu"%$#o" u"less $&ere #s '#ve" e6pl#%#$ly or #)pl#%#$ly ! do)!#" #.e. ! se$ of "u)bers o" 9&#%& $&e for)ul! $&e" def#"es ! fu"%$#o". He"%e $&e follo9#"' e6press#o"s !ppe!r: *e$ f6B L 67 be def#"e o" _ / 7 2`. *e$ '6B L s#" 6 be def#"ed for < ≤ 6 ≤ 76 Ho9ever #f $&e do)!#" of ! fu"%$#o" def#"ed by ! for)ul! #s $&e )!6#)u) se$ of re!l "u)bers for 9&#%& $&e for)ul! y#eld ! re!l "u)ber e.'. *e$ f6B L 1 for  ≠ < T&e" $&e do)!#" #s usu!lly "o$ s$!$ed e6pl#%#$ly. T&#s %o"ve"$#o" #s so)e$#)es %!lled $&e rule of $&e )!6#)u) do)!#". E-a%5#e 7.1<

Co"s#der $&e follo9#"' fu"%$#o"s 16B L 67 for 6 ≥ < 76B 16  7B for 6 ≠ < 36B L %os 6 for < ≤ 6 ≤ 7π 26B L $!" 6 for ≠ π7  "π "∈ 

T&e do)!#"s of f 7 !"d f 2 "eed "o$ &!ve bee" e6pl#%#$ly S$!$ed s#"%e e!%& %o"s#s$s of !ll $&ose "u)bers for (&#%& $&e for)ul! &!s )e!"#"' $&!$ #s $&e fu"%$#o"s Could &!ve bee" def#"ed by 9r#$#"' 16B L 67 !"d

f 26B L $!" 6 =>

MTH 131


Co"s#der $&e fu"%$#o" f6B L √1  67 F #$s do)!#" u"less o$&er9#se s$!$ed #s _ / 1 1`. Here 9e e6pl#%#$ly

E-a%5#e 7.<

Ass!%e tat te (o=do)!#" #s

3.9

ℜ.

Ca$a(te$"st"( F!n(t"ons

*e$ A be !"y subse$ of ! u"#vers!l se$ U. T&e" $&e re!l/v!lued fu"%$#o". A:U N1  < 1 #f 6 ∈ A 6A6B L < #f 6 ≠ A

def#"ed by

#s %!lled $&e %&!r!%$er#s$#% fu"%$#o" of A. E-a%5#e 8.1<

*e$ U L N! b % d e !"d A L Ns f e. T&e" $&e fu"%$#o" of U #"$o N1 < def#"ed by $&e follo9#"' d#!'r!) A 0 C D E

1 <

Is $&e %&!r!%$er#s$#% fu"%$#o" 6A of A o$e fur$&er $&!$ !"y fu"%$#o" f: U

N1 < def#"es ! subse$

Af L N6 6 ∈ U f6B L 1 Of U !"d $&!$ $&e %&!r!%$er#s$#% fu"%$#o"  Af of Af #s $&e or#'#"!l fu"%$#o" f. T&us $&ere #s ! o"e $o/o"e %orrespo"de"%e be$9ee" !ll subse$s of U #.e. $&e po9er se$ of U !"d $&e se$ of !ll fu"%$#o"s of U #"$o N1 <. 3.:

=?

O5e$at"ons

MTH 131


ou !re f!)#l#!r 9#$& $&e oper!$#o"s of !dd#$#o" !"d )ul$#pl#%!$#o" of "u)bers u"#o" !"d #"$erse%$#o" of se$s !"d %o)pos#$#o" of fu"%$#o"s. T&ese oper!$#o"s !re de"o$ed !s follo9s: A  b L % ! b L % A ∪ 0 L C A ∩ 0 L C 'o f L & I" e!%& s#$u!$#o" !" ele)e"$ % C or &B #s !ss#'"ed $o !" or#'#"!l p!#r of ele)e"$s. I" o$&er 9ords $&ere #s ! fu"%$#o" $&!$ !ss#'"s !" ele)e"$ $o e!%& ordered p!#r of ele)e"$s. (e "o9 #"$rodu%e De0"n"t"on :.1<

A" oper!$#o" α o" ! se$ A #s ! fu"%$#o" of $&e C!r$es#!" produ%$ A 6 A #"$o A #.e.

Re%a$* :.<

T&e oper!$#o" α A 6 A → A #s so)e$#)es referred $o !s ! &"na$, o5e$at"on !"d !" "/!ry oper!$#o" #s def#"ed $o be ! fu"%$#o" α : A @ A → A

(e s&!ll %o"$#"ue $o use $&e 9ord oper!$#o" #"s$e!d of b#"!ry oper!$#o" 3.:.1 C!%!#at")e O5e$at"ons

T&e oper!$#o" α: A 6 A →A #s %!lled %u)ul!$#ve #f for every ! b ∈ A α ! bB L α b !B

E-a%5#e 9.1<

Add#$#o" !"d )ul$#pl#%!$#o" of re!l "u)bers !re %u)ul!$#ve oper!$#o"s s#"%e A  b L  ! !"d

E-a%5#e 9.<

!b Lb!

ℜ be $&e oper!$#o" of *e$ α: ℜ 6 ℜ sub$r!%$#o" def#"ed by α: 6 yB 6y T&e" α81B L 2 !"d α1 8B L / 2 ==

MTH 131


He"%e sub$r!%$#o" #s "o$ ! %u)ul!$#ve oper!$#o"s s#"%e A ∪ 0 L 0 ∪ A

!"d

A ∩ 0 L 0 ∩ A

3.:. Asso("at")e O5e$at"ons

T&e oper!$#o" α: A 6 A → A #s %!lled !sso%#!$#ve #f for every ! b % ∈ A. αα! bB %B L α! αb %BB I" o$&er 9ords #f α! bB #s 9r#$$e" ! ∗ b $&e" α #s !sso%#!$#ve #f ! ∗ bB ∗ % L ! ∗ b ∗ %B E-a%5#e :.1<

Add#$#o" !"d )ul$#pl#%!$#o" of re!l "u)bers !re !sso%#!$#ve oper!$#o"s s#"%e !  bB  % L !  b  %B !"d

E-a%5#e :.<

!bB% L !b%B

*e$ α:ℜ 6 ℜ → ℜ be $&e oper!$#o" of d#v#s#o" def#"ed by α: 6 yB 6y T&e" α α17 4B 7B L α7 7B L 1 α17 α4 7BB L α17 3B L 2 He"%e d#v#s#o" #s "o$ !" !sso%#!$#ve oper!$#o".

E-a%5#e :.3<

U"#o" !"d #"$erse%$#o" of se$s !re !sso%#!$#ve oper!$#o"s s#"%e A ∪ 0 B ∪ C L A ∪ 0 ∪ CB 0B ∩ C L A ∩ 0 ∩ CB

3.:.3 D"st$"&!t")e O5e$at"ons

1<<

!"d A ∩

MTH 131


Co"s#der $&e follo9#"' $9o oper!$#o"s α: A 6 A → A β: A 6 A → A

T&e oper!$#o" α #s s!#d $o be d#s$r#bu$#ve over $&e oper!$#o" β #f for every ! b % ∈ A. α!βb %B L β bB α! %BB I" o$&er 9ords #f α! bB #s 9r#$$e" ! ∗ b. !"d β! bB #s 9r#$$e" ! ∆ b $&e" α d#s$r#bu$es over β #f

A ∗ b ∆ %B L ! ∗ bB ∆ ! ∗ %B E-a%5#e ;.1<

T&e oper!$#o" of )ul$#pl#%!$#o" of re!l "u)bers d#s$r#bu$es over $&e oper!$#o" of !dd#$#o" s#"%e Ab  %B L !b  !% 0u$ $&e oper!$#o" of !dd#$#o" of re!l "u)bers does "o$ d#s$r#bu$e over $&e oper!$#o" of )ul$#pl#%!$#o" s#"%e A  b%B ≠ !  bB!  %B

E-a%5#e ;.<

T&e oper!$#o" of U"#o" !"d #"$erse%$#o" of se$s d#s$r#bu$e over e!%& o$&er s#"%e A ∪ 0 ∩ CB L A ∪ 0B ∩ A ∪ CB A ∪ 0 ∪ CB L A ∩ 0B ∪ A ∩ CB

3.:.6 I+ent"t, E#e%ents

*e$ α: A 6 A A be !" oper!$#o" 9r#$$e" α! bB L ! ∗ b. A" ele)e"$ % ∈ A #s %!lled !" #de"$#$y ele)e"$ for $&e oper!$#o" α #f for every ! ∈ A. C ∗ ! L ! ∗ % L ! 1<1

MTH 131


E-a%5#e 1.1<

*e$ α:ℜ 6 ℜ → ℜ be $&e oper!$#o" of !dd#$# !dd#$#o". o". T&e" T&e" < #s !" #de"$#$y #de"$#$y ele)e ele)e"$ "$ for !dd#$#o" s#"%e for every re!l "u)ber ! ∈ℜ. < ∗ ! L ! ∗ < L ! $&!$ #s<  ! L !  < L !

E-a%5#e 1<

Co"s Co"s#d #der er $&e $&e op oper er!$ !$#o #o"" of #"$e #"$ers rse% e%$#$#o" o" of se$s se$s.. T&e" U $&e u"#vers!l se$ #s !" #de"$#$y ele)e"$ s#"%e for every se$ A 9&#%& #s ! subse$ of UB U ∗ A L A ∗ U L A)

$&!$ #s U ∩ A L A ∩ U L A

E-a%5#e 1.3<

Co"s#der $&e oper!$#o" of )ul$#pl#%!$#o" of Re!l "u)bers. "u)bers. T&e" $&e $&e "u)ber "u)ber 1 #s !" #de"$#$ #de"$#$yy ele)e"$ s#"%e for every re!l "u)ber ! 1 ∗ ! L ! ∗ 1 L !

Teo$e% :. 1<

$&!$ #s 1 • ! L ! • 1 L !

If !" oper!$ oper!$#o #o"" α: A 6 A → A &!s &!s !" #de"$ #de"$#$y #$y ele)e"$. T&us 9e %!" spe!+ of $&e #de"$#$y ele)e"$ for !" oper!$#o" #"s$e!d of !" #de"$#$y ele)e"$.

3.:. 3.:.7 7 In)e In)e$s $see E#e%e E#e%ent ntss

*e$ α: A 6 A → A be !" oper!$#o" 9r#$$e" α! bB L ! ∗ b !"d le$ e ∈ A be $&e #de"$#$y ele)e"$ for α. T&e" $&e #"verse of !" ele)e"$ ! ∈ A de"o$ed by !/1 #s !" ele)e"$ #" A 9#$& $&e follo9#"' proper$y: /1

E-a%5#e 11.1<

1<7

∗ ! L ! ∗ ! /1 L e

Co"s#der $&e oper!$#o" of !dd#$#o" of re!l "u)bers for 9&#%& < #s $&e #de"$#$y ele)e"$. ele)e"$. T&e" for !"y !"y re!l "u)ber "u)ber ! #$s "e'!$#ve "e'!$#ve / !B #s #$s !dd#$#ve !dd#$#ve #"verse s#"%e

MTH 131


/ ! ∗ ! L ! ∗ / ! L <)

E-a%5#e 11.

E-a%5#e 11.3

$&!$ #s / !B  ! L !  / !B L !  ! L<

Co"s#der $&e oper!$#o" of )ul$#pl#%!$#o" of r!$#o"!l "u)bers for 9&#%& 1 #s #s $&e #de"$#$y ele)e"$. T&e" for !"y "o"/ero r!$#o"!l "u)ber p 9&ere p !"d V  !re #"$e'ers #$s re%#pro%!l p #s #$s )ul$#pl#%!$#ve #"verse s#"%e pBpB L pBpB L 1 *e$ α:  6   be $&e oper!$#o" of )ul$#pl#%!$#o" for 9&#%& 1 #s $&e #de"$#$y ele)e"$F &ere  #s $&e $&e se$ of "!$ur!l "u)bers. T&e" 7 &!s "o )ul$#pl#%!$#ve #"verse s#"%e $&ere #s "o ele)e"$ 6 ∈  9#$& $&e proper$y  • 7 L 7 • 6 L 1 I" f!%$ "o ele)e"$ #"  &!s ! )ul$#pl#%!$#ve #"verse e6%ep$ 1 9&#%& &!s #$self !s !" #"verse.

3.;

O5e$at"ons An+ S! S!&sets

Co"s#der Co"s#der !" oper!$#o" oper!$#o" α: A 6 A → A !"d ! subse$ subse$ 0 of A. A. T&e" 0 #s s!#d $o be %losed u"der $&e oper!$#o" of α #f for every b bJ ∈ 0 $&!$ #s #f

αb bJB ∈ 0 α0 6 0B ⊂ 0

E-a%5#e1.1

Co"s#der $&e oper!$#o" of !dd#$#o" of "!$ur!l "u)bers. T&e" $&e se$ of eve" "u)bers #d %losed u"der $&e oper!$#o" of !dd#$#o" s#"%e $&e su) of !"y $9o eve" "u)bers #s !l9!ys eve". Moreover $&e se$ of odd "u)bers #s "o$ %losed u"der $&e oper!$#o" of !dd#$#o" s#"%e $&e su) of $9o odd "u)bers #s "o$ odd.

E-a%5#e 1.<

T&e four %o)ple6 "u)bers 1 / 1 I / I !re %losed 1<3

MTH 131


u"der $&e oper!$#o" of )ul$#pl#%!$#o". 6.

CONCLUSION

I" )!$& )!$&e) e)!$ !$#% #%!l !l !"!l !"!lys ys#s #s !"d !"d !bs$ !bs$r!% r!%$$ )!$& )!$&e) e)!$ !$#% #%s s $&e $&e u" u"#$#$ #s ! prereu#s#$e +"o9led'e. u"%$#o"s !"d d#!'r!)s 'o &!" #" &!"d. Se$ fu"%$# fu"%$#o"s o"s re!l/v! re!l/v!lue luedd fu"%$# fu"%$#o" o"s s %&!r!% %&!r!%$er $er#s$ #s$#% #% fu"%$# fu"%$#o"s o"s !re b!s#% b!s#% fu"%$#o"s #" !"!lys#s 7.

SUMMARY

See #f you re%!ll $&e follo9#"': T&e fu"% fu"%$#$#o" o" fJ #s %!ll %!lled ed ! res$ res$r# r#%$ %$#o #o"" of f $o $o 0 f f 0B of f: A → C #f '#ve" 0 ! subse$ of A f #"du%es ! fu"%$#o" ! fu"%$#o" fJ: 0 → C 9&#%& #s def#"ed by

/

JbB L fbB or !"y b∈ 0. / *e$ f: A → C !"d le$ 0 be ! superse$ of A. T&e" ! fu"%$#o" : 0 → C #s %!lled !" e6$e"s#o" of f #f for every ! ∈ A.

f!B L f!B / A fu"%$# fu"%$#o" o" #s #s %!lled %!lled ! se$ se$ fu"%$ fu"%$#o" #o" #f#f #$s #$s do)!#" do)!#" %o"s %o"s#s$ #s$ss of se$s se$s / A fu"%$# fu"%$#o" o" o" ! ! se$ se$ $&!$ $&!$ )!ps )!ps $&e $&e ele)e ele)e"$s "$s of of $&!$ $&!$ se$ #"$o #"$o $&e $&e re!l re!l "u)bers #s %!lled ! re!l/v!lued fu"%$#o". / T&e rule of M!6#)u) do)!#" #s used $o def#"e do)!#"s 9&#%& "eed

"o$ be s$!$ed e6pl#%#$ly s#"%e #$ #s $&e )!6#)u) se$ of re!l "u)bers for 9&#%& ! fu"%$#o" y#elds ! re!l "u)ber.

8.

TUTOR MAR MAR?E ?ED D ASSIGN SIGNME MEN NTS

1

*e$ ( L N! b b % dF v L N1 N1 7 3 3 !"d le$ f: ( → 5 be def#"ed by $&e !d,o#"#"' d#!'r!). d#!'r!). #"d: 1B fN! b dB 7B fN! %B.

1<2

MTH 131

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