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Kategori kesamaan
=
sama dengan
x = y berarti x and y mewakili hal atau
nilai yang sama.
1+1=2
umum Ketidaksamaan
≠
tidak sama dengan
x ≠ y berarti x dan y tidak mewakili hal
atau nilai yang sama.
1≠2
umum
< > ≤ ≥
ketidaksamaan lebih kecil dari lebih besar dari
x < y berarti x lebih kecil dari y. x > y means x lebih besar dari y.
3<4 5>4
!rder the!ry ine"uality x # y berarti x lebih kecil dari atau sama lebih kecil dari atau dengan y. sama dengan% lebih besar dari atau sama x $ y means x lebih besar dari atau sama dengan dengan y. !rder the!ry
3 # 4 and 5 # 5 5 $ 4 and 5 $ 5
tambah tambah
4 + & berarti 'umlah antara 4 dan &.
aritmatika dis'!int uni!n the dis'!int uni!n !* A1 + A2 means the dis'!int uni!n !* sets / and / A1 and A2. te!ri him0unan
!
2+(=) A1=1%2%3%4, ∧ A2=2%4%5%(, ⇒ A1 + A2 = -1%1% -2%1%
-3%1% -4%1% -2%2% -4%2% -5%2% -(%2,
kurang kurang
) 4 berarti ) dikurangi 4.
3=5
3 berarti negati* dari angka 3.
-5 = 5
aritmatika tanda negati*
negati* aritmatika setthe!retic c!m0lement minus with!ut
A B means the set that c!ntains all elements !* A that are n!t in B.
the
1%2%4, 1%3%4, = 2,
set the!ry multi0licati!n kali
3 4 means the multi0licati!n !* 3 by 4. ( = 5&
aritmatika artesian 0r!duct
"
the artesian 0r!duct !* / and / the direct 0r!duct !* / and /
X Y means the set !* all
!rdered 0airs with the *irst element !* each 0air selected *r!m 6 and the sec!nd element selected *r!m 7.
1%2, 3%4, = -1%3% -1%4%-2%3%-2%4,
te!ri him0unan cr!ss 0r!duct cr!ss
# $ means the cr!ss 0r!duct !* 8ect!rs -1%2%5 -3%4%1 = # and $ -22% 1&% 2
8ect!r algebra
%
di8isi!n 2 9 4 = .5
bagi & 9 3 atau &:3 berati & dibagi 3.
&
12:4 = 3
aritmatika s"uare r!!t akar kuadrat
'
; x berarti bilangan 0!siti* yang kuadratnya x.
;4 = 2
bilangan real c!m0le s"uare r!!t i* z = r e0-i is re0resented in 0!lar the c!m0le s"uare c!!rdinates with < # % then ; z = ;r ;-1 = i r!!t !* s"uare r!!t e0-i:2. bilangan c!m0le abs!lute 8alue
((
abs!lute 8alue !*
?x? means the distance in the
real line -!r ?3? = 3% ?5? = ?5? the c!m0le 0lane between x and @er!. ?i? = 1% ?3+4i? = 5
numbers *act!rial
nA is the 0r!duct 12... n.
4A = 1 2 3 4 = 24
) *
*akt!rial c!mbinat!rics 0r!bability distributi!n has distributi!n
X ~ D% means the rand!m 8ariable X has X ~ N(0,1), the standard normal distribution the 0r!bability distributi!n D.
statistika material im0licati!n A ⇒ B means i* A is true then B is als! true i* A is *alse then n!thing is said im0lies i* .. then ab!ut B.
+ 0r!0!siti!nal l!gic
B may mean the same as ⇒% !r it may ha8e the meaning *!r *uncti!ns gi8en bel!w.
x = 2 ⇒ x2 = 4 is true% but x2 = 4 ⇒ x = 2 is in general *alse -since x
c!uld be 2.
⊃ may mean the same as ⇒% !r it may
ha8e the meaning *!r su0erset gi8en bel!w. material e"ui8alence i* and !nly i* i**
, .
A ⇔ B means A is true i* B is true and A is *alse i* B is *alse.
x + 5 = y +2 ⇔ x + 3 = y
0r!0!siti!nal l!gic l!gical negati!n n!t 0r!0!siti!nal l!gic
Che statement D A is true i* and !nly i* A is *alse. E slash 0laced thr!ugh an!ther !0erat!r is the same as FDF 0laced in *r!nt.
l!gical c!n'uncti!n !r meet in a lattice and
Che statement A ∧ B is true i* A and B are b!th true else it is *alse.
0r!0!siti!nal l!gic% lattice the!ry l!gical dis'uncti!n !r join in a lattice !r 0r!0!siti!nal l!gic% lattice the!ry eclusi8e !r
Che statement A ∨ B is true i* A !r B -!r b!th are true i* b!th are *alse% the statement is *alse.
D-D A ⇔ A x ≠ y ⇔ D- x = y
n < 4 ∧ n >2 ⇔ n = 3 when n is a natural
number .
n $ 4 ∨ n # 2 ⇔ n ≠ 3 when n is a natural
number .
!r Che statement A ⊕ B is true when either -D A ⊕ A is always true% A B E !r G% but n!t b!th% are true. ⊻ 0r!0!siti!nal l!gic% A ⊕ A is always *alse. means the same. G!!lean algebra uni8ersal "uanti*icati!n *!r all *!r any *!r each
∀ xH P - x means P - x is true *!r all x.
∀ n ∈ NH n2 $ n.
0redicate l!gic eistential "uanti*icati!n there eists
∃ xH P - x means there is at least
!ne x
such that P - x is true.
∃ n ∈ NH n is e8en.
0redicate l!gic uni"ueness "uanti*icati!n
)
∃A xH P - x means there is eactly !ne x there eists eactly such that P - x is true. !ne
∃A n ∈
NH n + 5 = 2 n.
0redicate l!gic
/=
de*initi!n is de*ined as
x H= y !r x I y means x is de*ined t! be an!ther name *!r y -but n!te that I can
als! mean !ther things% such as c!ngruence.
0 e8erywhere
/
P H⇔ means P is de*ined t! be l!gically e"ui8alent t! .
c!sh x H= -1:2-e0 x + e0 - x A 6J B H⇔ - A ∨ B ∧ D- A ∧ B
set brackets
123
the set !* ...
a%b%!, means the set c!nsisting !* a% b% and !.
N = L%1%2%...,
te!ri him0unan
1/3
set builder n!tati!n
1(3
te!ri him0unan
the set !* ... such that ...
him0unan k!s!ng him0unan k!s!ng
x H P - x, means the set !* all x *!r n ∈ N H n2 < 2L, = which P - x is true. x ? P - x, is the same L%1%2%3%4, as x H P - x,. ∅ berarti him0unan yang tidak memiliki
elemen. , 'uga berarti hal yang sama.
n ∈ N H 1 < n2 < 4, = ∅
te!ri him0unan
set membershi0 is an element !* is n!t an element !*
a ∈ " means a is an element !* the set " a ∉ " means a is n!t an element !* " .
e8erywhere% te!ri him0unan
-1:21 ∈ N 21 ∉ N
subset is a subset !* te!ri him0unan
A ⊆ B means e8ery element !* A is als! element !* B.
A M B ⊆ A 4 ⊂ 5
A ⊂ B means A ⊆ B but A ≠ B.
su0erset is a su0erset !* te!ri him0unan
A ⊇ B means e8ery element !* B is als! element !* A.
A ∪ B ⊇ B 5 ⊃ 4
A ⊃ B means A ⊇ B but A ≠ B.
setthe!retic uni!n the uni!n !* ... and ... uni!n
A ∪ B means the set that c!ntains all the elements *r!m A and als! all th!se *r!m A ⊆ B ⇔ A ∪ B = B B% but n! !thers.
te!ri him0unan setthe!retic intersecti!n
6
intersected with intersect
A M B means the set that c!ntains all th!se elements that A and B ha8e in
c!mm!n.
x ∈ 5 H x2 = 1, M N = 1,
te!ri him0unan
7
setthe!retic c!m0lement minus with!ut
A N B means the set that c!ntains all th!se elements !* A that are n!t in B.
1%2%3%4, N 3%4%5%&, = 1%2,
te!ri him0unan
89
*uncti!n a00licati!n # - x berarti nilai *ungsi # 0ada elemen x. !* te!ri him0unan
Oika # - x H= x2% maka # -3 = 32 = ).
0recedence gr!u0ing
Per*!rm the !0erati!ns inside the 0arentheses *irst.
-:4:2 = 2:2 = 1% but : -4:2 = :2 = 4.
# H X B Y means the *uncti!n # ma0s the set X int! the set Y .
Qet # H : B N be de*ined by # - x = x2.
# ! $ is the *uncti!n% such that - # ! $ - x = # - $ - x.
i* # - x = 2 x% and $ - x = x + 3% then - # ! $ - x = 2- x + 3.
N means L%1%2%3%...,% but see the article !n natural numbers *!r a di**erent c!n8enti!n.
?a? H a ∈ :, = N
: means ...%3%2%1%L%1%2%3%...,.
a H ?a? ∈ N, = :
umum *uncti!n arr!w
f / X + *r!m ... t! Y te!ri him0unan o
*uncti!n c!m0!siti!n c!m0!sed with te!ri him0unan natural numbers
N
R numbers integers
:
S numbers rati!nal numbers
4
3.14 ∈ 4
T 4 means %:& H %%& ∈ :% & ≠ L,.
∉4
numbers real numbers
5
5 means limnBU an H ∀ n ∈ NH an ∈ 4% the limit eists,.
numbers
∈ 5 ;-1 ∉ 5
c!m0le numbers
C
C means a + bi H a%b ∈ 5 ,. numbers
i = ;-1 ∈ C
in*inity
;
in*inity numbers
U is an element !* the etended number line that is greater than all real numbers limBL 1:? x? = U it !*ten !ccurs in limits.
0i
0i
berarti 0erbandingan -rasi! antara keliling lingkaran dengan diameternya.
Wuclidean ge!metry
A = r V adalah luas
lingkaran dengan 'ari'ari -radius r
n!rm
(( ((
n!rm !* length !*
?? x?? is the n!rm !* the element x !* a n!rmed 8ect!r s0ace.
?? x+ y?? # ?? x?? + ?? y??
linear algebra summati!n
sum !8er ... *r!m ... X' =1n a' means a1 + a2 + ... + an. t! ... !*
X' =14 ' 2 = 12 + 22 + 32 + 42 = 1 + 4 + ) + 1& = 3L
aritmatika 0r!duct 0r!duct !8er ... *r!m ... t! ... !*
Y' =1n a' means a1a2ZZZan.
Y' =14 -' + 2 = -1 + 2-2 + 2-3 + 2-4 + 2 = 3 4 5 & = 3&L
Yi=LnY i means the set !* all -n+1tu0les - yL%...% yn.
Yn=135 = 5 n
aritmatika artesian 0r!duct the artesian 0r!duct !* the direct 0r!duct !* set the!ry deri8ati8e
?
# [- x is the deri8ati8e !* the *uncti!n # at / 0rime deri8ati8e the 0!int x% i.e.% the sl!0e !* the tangent \* # - x = x2% then # [- x = 2 x !* / there. kalkulus
inde*inite integral !r antideri8ati8e
@
inde*inite integral ] # - x d x means a *uncti!n wh!se !* / the deri8ati8e is # . antideri8ati8e !* /
] x2 d x = x3:3 +
kalkulus de*inite integral
]Lb 2 d x = b3:3
integral *r!m ... t! ... !* ... with res0ect t! kalkulus
]ab # - x d x means the signed area between the xais and the gra0h !* the *uncti!n # between x = a and x = b.
gradient del% nabla% gradient ∇ # -1% /% n is the 8ect!r !* 0artial !* deri8ati8es -d# : dx1% /% d# : dxn.
\* # - x% y% z = 3 xy + z V then ∇ # = -3 y% 3 x% 2 z
kalkulus 0artial deri8ati8e
A
^ith # -1% /% n% _*:_i is the deri8ati8e \* # -%y = 2y% then _ # :_ 0artial deri8ati8e !* !* # with res0ect t! i% with all !ther = 2y 8ariables ke0t c!nstant. kalkulus b!undary b!undary !*
_ means the b!undary !*
_ H ???? # 2, = H ?? ?? = 2,
x ⊥ y means x is 0er0endicular t! y !r m!re generally x is !rth!g!nal t! y.
\* l ⊥m and m⊥n then l ?? n.
t!0!l!gy 0er0endicular is 0er0endicular t! ge!metri b!tt!m element the b!tt!m element x = ⊥ means x is the smallest element.
∀ x H x ∧ ⊥ = ⊥
lattice the!ry entailment
(=
entails m!del the!ry
A ⊧ B means the sentence A entails the sentence B% that is e8ery m!del in which A ⊧ A ∨ D A A is true% B is als! true.
in*erence
(B
in*ers !r is deri8ed *r!m x ⊢ y means y is deri8ed *r!m x.
A B B ⊢ D B B D A
0r!0!siti!nal l!gic% 0redicate l!gic n!rmal subgr!u0 ◅
is a n!rmal subgr!u0 !*
N means that N is a n!rmal subgr!u0 * - !* gr!u0 . ◅
gr!u0 the!ry
&
"u!tient gr!u0 m!d
: + means the "u!tient !* gr!u0 m!dul! its subgr!u0 + .
◅
gr!u0 the!ry is!m!r0hism
is is!m!r0hic t! gr!u0 the!ry
L% a% 2a% b% b+a% b+2a, : L% b, = L% b,% a% : 1% 1, ` % ` + means that gr!u0 is is!m!r0hic where is the "uaterni!n t! gr!u0 + gr!u0 and is the Klein *!urgr!u0.
