Exercise Set 4.1

NVNTLO\N \NP 0-7

<-

Qn i~|p ﬇m` m~idnt| l7# l4# l<# gm` l0 |~lf pfgp l7&“7# <# 4# 2/ * l4&4# 2# 0# “7/ * l<&1# 7# 7# 0/ * l0&5# <# 7# 4/ 9 &2# ># 5# “
* l< *

4 l7 * 0 l4 *

l< * l0 9 5

“l4 * 0l< * 4l 4 l0 9 < Pfn g~ainmpn` igptov he pfo| |y|pni o|

 ∔7 4  < 2   4 0   2 ∔7

1

5

7

<

7

7

0

4

  > 5  ∔<  2

Pfn tn`~ln` thq)nlfnchm ehti he pfo| igptov o|

7 2  2  2

2

2

2

7

7

2

2

7

2

7

2

2

2

7

Pf~| l7 9 7# l4 9 7# 7 # l< 9 “7# gm` l0 9 7-

777



∔7  7

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774

Nvntlo|n \np 0-7

>-

&l/  u 9 Z<4 * 0 4 * 2 4 * &“74/4Y

?-

&g/ &4#>/

․

75? 9 7< ¢¢¢¢¢¢

9

&“0#/&
&l/ &<# 7# 0# “>/

77-

7(4

․

&4# 4# “0# “ 9 1

&g/ `&~# u u/ 9 Z&7 “ 4/4 * &“4 “ 7/4Y

7(4

9

72 ¢¢¢¢¢

&l/ `&~# u/ 9 Z&2 *
7>-

7(4

9

>? ¢¢¢¢¢

&g/ Qn chhj eht ugc~n| he j |~lf pfgp ~

․

Lcngtcy j 9 “< o| pfn hmcy rh||odcopy-

71-

&g/ Qn fgun s~

․

us

9 s<&0/ * 4&“7/s 9 72# qfocn ~  u 9 Z< 4 * 4 4Y7(4Z04 * &“7/4Y7(4 9

&`/ Fntn s~

․

us

447¢¢¢¢¢¢

9 2 * 4 * 4 * 7 9 ># qfocn

~  u 9 Z2 4 * &“4/4 * 4 4 * 7 4Y7(4Z&“7/4 * &“7/4 * 7 4 * 7 4Y7(4 9 5-

4<-

Qn i~|p |nn oe pfn |y|pni < * 0p 0 p 9 | 4 * 5p 5 p 9 < “ < | < * 0p 0 p 9 > “ 0 | “7 “ 4p 4p 9 0 “ 4 | o| lhm|o|pnmp- \hcuoma pfn ﬇t|p pqh nw~gpohm| yonc` p 9 “ 0(?# | 0(?# | 9 77(?- \~d|pop~poma omph pfn (< 9 7(?- Pf~| pfn |y|pni o| omlhm|o|pnmp# omlhm|o|pnmp # |h pfn comn| gtn |jnq |jnq-









Nvntlo|n \np 0-7

41-

77<

Cnp ~ 9 &~ & ~7# ‥# ~ m/# u 9 & u7# ‥# u m/# gm` q 9 & q7# ‥# q ‥# q m/-

&g/ ~

․

&j u/ 9 &~7# ‥# ~ m/

․

&ju7# ‥# ju m/

9 ~7 ju7 * ‥ * ~ m ju m 9 j&~7 u7 * ‥ * ~ m u m/ 9 j& ~

&d/ &d/ ~

․

u/

․

& u * q/ 9 &~7# ‥# ~ m/

․

& u7 * q7# ‥# u ‥# u m * q m/

9 ~7& u7 * q7/ * ‥ * ~ m & u m * q m/ 9 &~ & ~7 u7 * ‥ * ~ m u m/ * &~7 q7 * ‥ * ~ m q m/ 9~

․

u*~

․

q

<>-

&g/ Dy pfnhtni 0-7-1# qn fgun `&~# u /

<1-

&g/ Pt~n- Om anmntgc# qn jmhq pfgp

9

~“u

9

~

4

*

u

4

9

4-

~ * u4 9 ~4 *  u4 * 4& ~ ․ u/ \h om pfo| lg|n ~

․

u 9 2 gm` pfn unlpht| gtn htpfhahmgc-

t~n- Qn gtn gtn aounm aounm pfgp pfgp ~ ․ u 9 ~ ․ q 9 2- D~p |omln ~ &d/ Pt~nehcchq| pfgp ~ o| htpfhahmgc ph u * q-

․

& u * q/ 9 ~

․

u *~

․

q# op

hdpgom g lh~mpntnvgircn# lh~mpntnvgircn# cnp ~ 9 &7# 2# 2/# u 9 &7# 7# 2/# gm` q 9 &“7# 7# 2/&l/ Egc|n- Ph hdpgom

















NVNTLO\N \NP 0-4

7-

&d/ \omln pfn ptgm|ehtigpohm igr| & v7# v4/ ph & q7# q4# q
<-

Pfn |pgm`gt` igptov o| G# G# qfntn

q9

 < > ∔7   v7     Gv 9  0 ∔7 7   v4   < 4 ∔7   v< 

|h pfgp

 < > ∔7   ∔7   <        P & ∔7# 4# 0 / 9  0 ∔7 7   4  9  ∔4   < 4 ∔7   0   ∔< 

>-

&g/ Pfn |pgm`gt` igptov o|

 2 7  ∔7 2     7 <    7 ∔7  Mhpn pfgp P &7# &7# 2/ 9 &2# “7# 7# 7/ gm` P &2# &2# 7/ 9 &7# 2# <# “7/-









775

1-

Nvntlo|n \np 0-4

&d/ Fntn

 4 ∔7  P & 4# 7 ∔
?-

  7  2  7

 4  79    ∔< 

 2  ∔4     2 

&g/ Om pfo| lg|n#

7  P & 4# ∔ >#
2

2

 4  ∔>  9    < 

 2 ∔7

7 2

 4  ∔>     ∔< 

|h pfn tnfflnlpohm he &4# “># # “
7<-

&d/ Pfn oigan he &“4# 7# 4/ o| &2# 7# 4

 lh| & 0>² /  2   ∔ |om & 0>² / 

7>-

2 7 2

  2   lh| & 0>² /  |om & 0>² /

  ∔4    79     4   ∔ 

oigan he &“4# &“4# 7# 4/ 4/ o| &2# 7# 4 &d/ Pfn oigan

 lh| & ∔0>² /  2    |om & ∔0>² /

2 7 2

4/# |omln ¢¢¢¢

∔ |om & ∔0>² /   2   lh| & ∔0>² /     9  ∔ 

7 4 2 7

7

2

4 2

7

7

2

4

      

 ∔4   79    4 

 ∔4   7 9    4

 2    7   4 4   

7 4 2 7 4

4/# |omln |omln ¢¢¢¢

 ∔4   7    4  2 7 2

  4 2   7   7

 2     7  4 4   











Nvntlo|n \np 0-4

71-

771

&g/ Pfn |pgm`gt` igptov o|

2 7 

 7  2 2

7

 2   2   

7

<

∔

4

4

<

7

4

4

  9   

2  7  4

2 <

∔

4

   

|pgm`gt` gt` igpto igptov v eht g lh~mpntl lh~mpntlchlj chljqo| qo|n n thpgpohm thpgpohm he 7>² 7>² * 72>² 72>² * 52² 9 762² o| &l/ Pfn |pgm`

 lh| & 762² / ∔ |om & 762² /   ∔7 2   9  ² ² | o m 7 6 2 l h | 7 6 2 & / & /    2 ∔7 

7?-

&l/ Pfn |pgm`gt` igptov o|

lh| &762²/ ∔ |om &762²/   |om &762²/ lh| &762²/  2  2

2

 2  7

 ∔7 2 9  2 ∔7  2 2  2 9  2  ∔7

7 2 2

 lh| & ?2²/  2   ∔ |om & ?2²/  2 2 7

   

2

∔7  2 

 2  2   ∔7

2 7 2

2

7

7

2

2

2

|om & ?2² / 

 2  lh| h| & ?2² /      

   

7  2 2 

7

2

2

2

2

7

2

∔7

2

  lh| & 412² / ∔ |om & 412²/   |om & 412² / lh| & 412² /  2

   

2









776

Nvntlo|n \np 0-4

4<-

\np &g &g# d# l/ nw~gc ph &7# 2# 2/# &2# 7# 2/# gm` &2# 2# 7/ om p~tm-

4>-

&g/ \omln P 4&P 7& v7# v4// 9 &<& v7 * v4/# 4& v7 * v4/ * 0& v7 “ v4// 9 &< v7 * < v4# 5 v7 “ 4 v4/# qn fgun ZP4



< <   5 ∔4 

P 7 Y 9 

Qn gc|h fgun

< 4

ZP4 Y ZP 7 Y 9 

2  7



0  7

7

< < 9 ∔7   5 ∔4 

41-

Lhir~pn pfn ptgln he pfn igptov aounm om Ehti~cg &71/ gm` ~|n pfn eglp pfgp & g# d# l/ o| g ~mop unlpht-

4?-

v)gvo| gm` g òcgpohm dy g eglpht he 4&g/ Pfo| o| gm htpfhahmgc rthknlpohm hm pfn v)gvo| v)gvo| gm` g òcgpohm dy g eglpht he 4&d/ Pfo| o| g tnfflnlpohm gdh~p pfn v)gvo|

<7-

\omln lh|&4 ζ / 9 lh| 4 ζ “ |om 4 pfth~af gm gmacn he 4 ζ -

ζ

gm` |om&4ζ / 9 4 |om ζ lh| ζ # pfo| tnrtn|nmp| g thpgpohm









NVNTLO\N \NP 0-<

7-

&g/ Rthknlpohm| gtn mhp hmn)ph)hmn |omln pqh ò|pomlp unlpht| lgm fgun pfn |gin oigan unlphthe T T4 ht T ht T< hmph op|nce&d/ \omln g tnfflnlpohm o| op| hqm omunt|n# op o| g hmn)ph)hmn igrroma he

<-

Oe qn tn`~ln pfn |y|pni he nw~gpohm| ph thq)nlfnchm ehti# qn ﬇m` pfgp q7 9 4 q4# |h pfgp gmy unlpht om pfn tgman i~|p dn he pfn ehti &4 q# q# q/ q/- Pf~| &<# 7/# eht nvgircn# o| mhp om pfn tgman-

>-

&g/ \omln pfn `npntiomgmp he pfn igptov ZP Y

 7 9   ∔7

4



7

o| <# pfn ptgm|ehtigpohm P o| P o| hmn)ph)hmn qopf

7 < 7 ZP Y ∔ 9  7  < Pf~| P

7

&

/

 7

4

7

4 ∔  <  7  <  7











742

?-

Nvntlo|n \np 0-<

P o| comngt |omln &g/ P o| P &&& & v7# y7/ * & v4# y4/ / 9 &4& v7 * v4/ * &y7 * y4/# & v7 * v4/ “ &y & y7 * y4// 9 &4 v7 * y7# v7 “ y7/ * &4 v4 * y4# v4 “ y4/ 9 P & v7# y7/ * P & v4# y4/

gm` P &j& v# v# y// 9 &4jv &4jv * jy# jy# jv “ jy/ jy/ 9 j&4 v * y# v “ y/ 9 jP & v# v# y/

&d/ \omln P && && v7# y7/ * & v4# y4//9 & v7 * v4 * 7# y7 * y4/ 9 & v7 * 7# y7/ * & v4# y4/

≢ P & v7# y7/ * P & v4# y4/ gm` P &j& v# v# y// 9 &jv & jv * 7# jy/ jy/

7<-

jP & v# v# y/ ~mcn|| j 9 7# ≢ jP &

P o| P o| mhmcomngt-

&g/ Pfn rthknlpohm |nm`| n7 ph op|nce gm` pfn tnfflnlpohm |nm`| n7 ph “n7# qfocn pfn rthknlpohm |nm`| n4 ph pfn {nth unlpht# qfolf tnigom| ﬇vn` ~m`nt pfn tnfflnlpohm- Pfntnehtn

 ∔7  2

P &n7/ 9 &“7# 2/ gm` P &n4/ 9 &2# 2/# |h pfgp ZP Y 9 

2



2









Nvntlo|n \np 0-<

71-

747

&g/ Dy pfn tn|~cp he Nvgircn >#

  ∔7      9   4     7 &

P 

&7 4

4

4

/

/& 7

&7 4

/

4

&7

/& 7 4

4

/

4

/   ∔7  9  7 4    

ht P &“7# &“7# 4/ 9 &7(4# 7(4/

7?-

&g/

 ∔7 G 9  2  2 

Noanmugc~n

Noanmugc~n

&d/

 7 G 9  2   2

κ 7 9 2#

κ

2 

2 7

2 

2

7  



κ 7 9 “7# noanmunlpht

κ 4 9 7# noanmunlpht

2

2 

2

 7  

2

 2   2    ν47 9  7  # ν44 9  2    2     7  

2 

ν7 9  

ν

 7 ν7 9  2   2 

2 7 2

  

 7    ν

 2   

 |   



4

7 4   











744

Nvntlo|n \np 0-<

ht &κ “ 4/ < 9 2- Pf~| pfn hmcy noanmugc~n o| noanmunlpht# pfnm

   

2

2

2

2

2

2

2

2

2

   

 v y   {

 9  

κ 9    

2 2 2

4- Oe & v# v# y# {/ {/ o| g lhttn|rhmòma

   

\omln pfn gdhun nw~gpohm fhc`| eht nunty unlpht & v & v## y# {/# {/# nunty mhm{nth unlpht o| gm noanmunlpht {)gvo| ~mlfgman` gm` gcpnt| &d~p &`/ \omln pfn ptgm|ehtigpohm cngun| gcc unlpht| hm pfn {)gvo| `hn| mhp tnunt|n/ pfn òtnlpohm he gcc hpfnt unlpht|# op| hmcy noanmugc~n o| κ 9 7 qopf lhttn|rhmòma noanmunlpht| &2# 2# {/ {/ qopf { ≢ 2- Ph untoey pfo|# hd|ntun pfgp pfn lfgtglpnto|pol nw~gpohm o|

κ∔7 4 ∔7 4

κ∔7 4

2

2

2

κ ∔7 ∔7

7

4

2

92

ht

& κ ∔ 7/

κ ∔7 7

4 4

7

κ ∔7

4 4

9 & κ ∔ 7/ & κ ∔ 7 

4

4

/

* &7

4

/

4

92 









Nvntlo|n \np 0-<

4<-

74<

Ethi Eoa~tn 7# qn |nn pfgp P &n7/ 9 &lh| 4 ζ # |om 4ζ / gm` ethi Eoa~tn 4# pfgp P &n4/ 9

  <ς   <ς   # |om  * 4ζ   9 &|om 4ζζ # ∔ lh| 4ζ /  4 * 4ζ   lh|    4   y &2# 7/ ς

<ς

y

4

4

*

∔ ζ

4ζ

ζ

&lh| 4ζ # |om 4ζ /

v ς

4

c

ζ

c

∔ ζ

4ζ ζ

v &7# 2/

Eoa~tn 7

  <ς  *  lh|   4 Eoa~tn 4

 

 <ς  4 *

4ζ  # |om 

   

4ζ  

















NVNTLO\N \NP 0-0

7-

&g/ & v4 * 4 v “ 7/ “ 4&< v4 * 4/ 9 “> v4 * 4 v “ > v/ * 5& v4 * 4 v * 4/ 9 45 v4 * 41 v * 5 &d/ >(0 v4 * < v/ v/ 9 v0 * v4 * 7 &l/ & v0 * 4 v< * v4 “ 4 v * 7/ “ &4 v< “ 4 v/

&`/

<-

ς&0 v< “ < v4 * 1 v * 7/ 9 0 ς v< “ < ς v4 * 1 ς v * ς

igrroma e  T< &g/ Mhpn pfgp pfn igrroma e

dy e &g# d# l/ 9 sgs fg| e fg| e &7# &7# 2# 2/ 9 7# e 7# e &2# &2# 7# 2/ 9 ↔ T aounm dy e

2# gm` e gm` e &2# &2# 2# 7/ 9 2- \h Oe e qntn e qntn g comngt igrroma# pfn igptov qh~c` dn G 9 &7# 2# 2/-

 ∔7 









745

1-

Nvntlo|n \np 0-0

& g “ d/ &g/ P &gv * d/ 9 &g * d/ v * &g P  R7

↔ R7

P &gv * d/ 9 gv4 * &g & g * d/ v * &4g &4 g “ d/ &d/ P & P  R7

↔ R4

4 l “ `/ v * &4g &4 g * d * l * <` < `/ &l/ P &gv< * dv4 * lv * `/ 9 &g * 4l P  R<

↔ R7

&`/ P &gv4 * dv * l/ 9 dv P  R4

↔ R4

&n/ P &gv4 * dv * l/ 9 d P  R4 ↔ R2

?-

< n“p &g/
 2

7

2

2 











Nvntlo|n \np 0-0

741

&d/ Pfn |y|pni o|;

 7 7  7 

2 4 <

2   d2

  7  2   d7  9  7        d  <       0  4

\hcuoma; d2 9 7# d7 9 2# d4 9 7Pf~|# r Pf~|# r& & v/ v/ 9 7

․

& v * 4/& v/ v/ * 2

9 & v * 4/

․

v*7

9 v4 * 4 v * 7

․

& v * 4/ * 7 9 & v * 4/

․

v*7









746

Nvntlo|n \np 0-0

&`/ Qn fgun

71-

 d2   7 ∔4 0 ∔6   4   ∔72   d   2 7 ∔< 1   ∔4   74   7  9     9    d4   2 2 7 ∔4   2   ∔0      4   4   2 2 2 7       d<

&g/

 g2   7 ∔ v2   d2  9  g   d   2 7     7  7 

&d/

 g2   7 ∔ v2  g  9  2 7  7   2 2  g     4

  d2   d  ∔& v2 * v7 /   7     d  7    v2 v7

4









Nvntlo|n \np 0-0

47-

74?

mhpn R 9 yo- Fnmln pfn Ugm`ntihm`n |y|pni fg| g ~mow~n |hc~pohm pfgp gtn pfn &g/ Qn ﬇t|p mhpn R lhne﬇lonmp| he pfn rhcymhiogc he m he mpf pf `natnn pfth~af pfn m pfn m * 7 `gpg rhomp|- \h# pfntn o| nvglpcy hmn rhcymhiogc he pfn mpf mpf `natnn pfth~af m * 7 `gpg rhomp| qopf pfn vo ~mow~n- Pf~|# pfn Cgatgman nvrtn||ohm i~|p dn gcandtgolgccy nw~ougcnmp ph pfn Ugm`ntihm`n ehti2# 7# 7# ‥# m ‥# m## pfnm pfn comngt |y|pni| eht pfn Ugm`ntihm`n gm` Mnqphm| &d/ \omln lo 9 yo# o 9 2# inpfh` tnigom pfn |gin-

&l/ Mnqphm‟| ehti gcchq| eht pfn ng|y g`òpohm he gmhpfnt rhomp & v & v m*7 # y m*7 / pfgp `hn| mhp m*7 m*7 fgun ph dn om gmy ht`nt qopf tn|rnlp ph pfn hpfnt vo ugc~n|- Pfo| o| `hmn dy g`òma g mnvp pnti ph r ph r&&o/# r /# rn r m*7 & v/ v/ 9 d m*7 & v “ v2/& v “ v7/& v “ v4/ ‥ & v “ v m/ m*7 m*7 * d m& v “ v2/& v “ v7/& v “ v4/ ‥ & v “ v m“7 / m“7

Exercise Set 4.1

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