Simbol matematika dasar

Simbol
 Nama Penjelasan Contoh
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Kategori
=
 Kesamaan x = y berarti x and y mewakili hal atau nilai yang sama. 1 + 1 = 2
sama dengan
umum
≠
 Ketidaksamaan x ≠ y berarti x dan y tidak mewakili hal atau nilai yang sama. 1 ≠ 2
tidak sama dengan
umum
<

>
 Ketidaksamaan x < y berarti x lebih kecil dari y.

x > y means x lebih besar dari y.		 3 < 4
5 > 4
lebih kecil dari; lebih besar dari
order theory
≤

≥
 Ketidaksamaan x ≤ y berarti x lebih kecil dari atau sama dengan y.

x ≥ y berarti x lebih besar dari atau sama dengan y.		 3 ≤ 4 and 5 ≤ 5
5 ≥ 4 and 5 ≥ 5
lebih kecil dari atau sama dengan, lebih besar dari atau sama dengan
order theory
+
 Perjumlahan 4 + 6 berarti jumlah antara 4 dan 6. 2 + 7 = 9
tambah
aritmatika
disjoint union A1 + A2 means the disjoint union of sets A1 and A2. A1={1,2,3,4} ∧ A2={2,4,5,7} ⇒
A1 + A2 = {(1,1), (2,1), (3,1), (4,1), (2,2), (4,2), (5,2), (7,2)}
the disjoint union of … and …
teori himpunan
−
 Perkurangan 9 − 4 berarti 9 dikurangi 4. 8 − 3 = 5
kurang
aritmatika
tanda negatif −3 berarti negatif dari angka 3. −(−5) = 5
negatif
aritmatika
set-theoretic complement A − B berarti himpunan yang mempunyai semua anggota dari A yang tidak terdapat pada B. {1,2,4} − {1,3,4}  =  {2}
minus; without
set theory
×
 multiplication 3 × 4 berarti perkalian 3 oleh 4. 7 × 8 = 56
kali
aritmatika
Cartesian product X×Y means the set of all ordered pairs with the first element of each pair selected from X and the second element selected from Y. {1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)}
the Cartesian product of … and …; the direct product of … and …
teori himpunan
cross product u × v means the cross product of vectors u and v (1,2,5) × (3,4,−1) =
(−22, 16, − 2)
cross
vector algebra
÷

/
 division 6 ÷ 3 atau 6/3 berati 6 dibagi 3. 2 ÷ 4 = .5

12/4 = 3
bagi
aritmatika
√
 square root √x berarti bilangan positif yang kuadratnya x. √4 = 2
akar kuadrat
bilangan real
complex square root if z = r exp(iφ) is represented in polar coordinates with -π < φ ≤ π, then √z = √r exp(iφ/2). √(-1) = i
the complex square root of; square root
Bilangan kompleks
| |
 absolute value |x| means the distance in the real line (or the complex plane) between x and zero. |3| = 3, |-5| = |5|
|i| = 1, |3+4i| = 5
nilai mutlak dari
numbers
!
 factorial n! adalah hasil dari 1×2×...×n. 4! = 1 × 2 × 3 × 4 = 24
faktorial
combinatorics
~
 probability distribution X ~ D, means the random variable X has the probability distribution D. X ~ N(0,1), the standard normal distribution
has distribution; tidk terhingga
statistika
⇒

→

⊃
 material implication A ⇒ B means if A is true then B is also true; if A is false then nothing is said about B.

→ may mean the same as ⇒, or it may have the meaning for functions given below.

⊃ may mean the same as ⇒, or it may have the meaning for superset given below.		 x = 2  ⇒  x2 = 4 is true, but x2 = 4   ⇒  x = 2 is in general false (since x could be −2).
implies; if .. then
propositional logic
⇔

↔
 material equivalence A ⇔ B means A is true if B is true and A is false if B is false. x + 5 = y +2  ⇔  x + 3 = y
if and only if; iff
propositional logic
¬

˜
 logical negation The statement ¬A is true if and only if A is false.

A slash placed through another operator is the same as "¬" placed in front.		 ¬(¬A) ⇔ A
x ≠ y  ⇔  ¬(x =  y)
not
propositional logic
∧
 logical conjunction or meet in a lattice The statement A ∧ B is true if A and B are both true; else it is false. n < 4  ∧  n >2  ⇔  n = 3 when n is a natural number.
and
propositional logic, lattice theory
∨
 logical disjunction or join in a lattice The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false. n ≥ 4  ∨  n ≤ 2  ⇔ n ≠ 3 when n is a natural number.
\
propositional logic, lattice theory

⊕


⊻
||exclusive or		 The statement A ⊕ B is true when either A or B, but not both, are true. A ⊻ B means the same. (¬A) ⊕ A is always true, A ⊕ A is always false.
xor
propositional logic, Boolean algebra
∀
 universal quantification ∀ x: P(x) means P(x) is true for all x. ∀ n ∈ N: n2 ≥ n.
for all; for any; for each
predicate logic
∃
 existential quantification ∃ x: P(x) means there is at least one x such that P(x) is true. ∃ n ∈ N: n is even.
there exists
predicate logic
∃!
 uniqueness quantification ∃! x: P(x) means there is exactly one x such that P(x) is true. ∃! n ∈ N: n + 5 = 2n.
there exists exactly one
predicate logic
:=

≡

:⇔
 definition x := y or x ≡ y means x is defined to be another name for y (but note that ≡ can also mean other things, such as congruence).

P :⇔ Q means P is defined to be logically equivalent to Q.		 cosh x := (1/2)(exp x + exp (−x))

A XOR B :⇔ (A ∨ B) ∧ ¬(A ∧ B)
is defined as
everywhere
{ , }
 set brackets {a,b,c} means the set consisting of a, b, and c. N = {0,1,2,...}
the set of ...
teori himpunan
{ : }

{ | }
 set builder notation {x : P(x)} means the set of all x for which P(x) is true. {x | P(x)} is the same as {x : P(x)}. {n ∈ N : n2 < 20} = {0,1,2,3,4}
the set of ... such that ...
teori himpunan

∅

{}
 himpunan kosong ∅ berarti himpunan yang tidak memiliki elemen. {} juga berarti hal yang sama. {n ∈ N : 1 < n2 < 4} = ∅
himpunan kosong
teori himpunan
∈

∉
 set membership a ∈ S means a is an element of the set S; a ∉ S means a is not an element of S. (1/2)−1 ∈ N

2−1 ∉ N
is an element of; is not an element of
everywhere, teori himpunan
⊆

⊂
 subset A ⊆ B means every element of A is also element of B.

A ⊂ B means A ⊆ B but A ≠ B.		 A ∩ B ⊆ A; Q ⊂ R
is a subset of
teori himpunan
⊇

⊃
 superset A ⊇ B means every element of B is also element of A.

A ⊃ B means A ⊇ B but A ≠ B.		 A ∪ B ⊇ B; R ⊃ Q
is a superset of
teori himpunan
∪
 set-theoretic union A ∪ B means the set that contains all the elements from A and also all those from B, but no others. A ⊆ B  ⇔  A ∪ B = B
the union of ... and ...; union
teori himpunan
∩
 set-theoretic intersection A ∩ B means the set that contains all those elements that A and B have in common. {x ∈ R : x2 = 1} ∩ N = {1}
intersected with; intersect
teori himpunan
\
 set-theoretic complement A \ B means the set that contains all those elements of A that are not in B. {1,2,3,4} \ {3,4,5,6} = {1,2}
minus; without
teori himpunan
( )
 function application f(x) berarti nilai fungsi f pada elemen x. Jika f(x) := x2, maka f(3) = 32 = 9.
of
teori himpunan
precedence grouping Perform the operations inside the parentheses first. (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.

umum
f:X→Y
 function arrow f: X → Y means the function f maps the set X into the set Y. Let f: Z → N be defined by f(x) = x2.
from ... to
teori himpunan
o
 function composition fog is the function, such that (fog)(x) = f(g(x)). if f(x) = 2x, and g(x) = x + 3, then (fog)(x) = 2(x + 3).
composed with
teori himpunan

N

ℕ
 Bilangan asli N berarti {0,1,2,3,...}, but see the article on natural numbers for a different convention. {|a| : a ∈ Z} = N
N
Bilangan

Z

ℤ
 Bilangan bulat Z berarti {...,−3,−2,−1,0,1,2,3,...}. {a : |a| ∈ N} = Z
Z
Bilangan

Q

ℚ
 Bilangan rasional Q berarti {p/q : p,q ∈ Z, q ≠ 0}. 3.14 ∈ Q

π ∉ Q
Q
Bilangan

R

ℝ
 Bilangan real R berarti {limn→∞ an : ∀ n ∈ N: an ∈ Q, the limit exists}. π ∈ R

√(−1) ∉ R
R
Bilangan

C

ℂ
 Bilangan kompleks C means {a + bi : a,b ∈ R}. i = √(−1) ∈ C
C
Bilangan
∞
 infinity ∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits. limx→0 1/|x| = ∞
infinity
numbers
π
 pi π berarti perbandingan (rasio) antara keliling lingkaran dengan diameternya. A = πr² adalah luas lingkaran dengan jari-jari (radius) r
pi
Euclidean geometry
|| ||
 norm ||x|| is the norm of the element x of a normed vector space. ||x+y|| ≤ ||x|| + ||y||
norm of; length of
linear algebra
∑
 summation ∑k=1n ak means a1 + a2 + ... + an. ∑k=14 k2 = 12 + 22 + 32 + 42 = 1 + 4 + 9 + 16 = 30
sum over ... from ... to ... of
aritmatika
∏
 product ∏k=1n ak means a1a2···an. ∏k=14 (k + 2) = (1  + 2)(2 + 2)(3 + 2)(4 + 2) = 3 × 4 × 5 × 6 = 360
product over ... from ... to ... of
aritmatika
Cartesian product ∏i=0nYi means the set of all (n+1)-tuples (y0,...,yn). ∏n=13R = Rn
the Cartesian product of; the direct product of
set theory
'
 derivative f '(x) is the derivative of the function f at the point x, i.e., the slope of the tangent there. If f(x) = x2, then f '(x) = 2x
… prime; derivative of …
kalkulus
∫
 indefinite integral or antiderivative ∫ f(x) dx means a function whose derivative is f. ∫x2 dx = x3/3 + C
indefinite integral of …; the antiderivative of …
kalkulus
definite integral ∫ab f(x) dx means the signed area between the x-axis and the graph of the function f between x = a and x = b. ∫0b x2  dx = b3/3;
integral from ... to ... of ... with respect to
kalkulus
∇
 gradient ∇f (x1, …, xn) is the vector of partial derivatives (df / dx1, …, df / dxn). If f (x,y,z) = 3xy + z² then ∇f = (3y, 3x, 2z)
del, nabla, gradient of
kalkulus
∂
 partial derivative With f (x1, …, xn), ∂f/∂xi is the derivative of f with respect to xi, with all other variables kept constant. If f(x,y) = x2y, then ∂f/∂x = 2xy
partial derivative of
kalkulus
boundary ∂M means the boundary of M ∂{x : ||x|| ≤ 2} =
{x : || x || = 2}
boundary of
topology
⊥
 perpendicular x ⊥ y means x is perpendicular to y; or more generally x is orthogonal to y. If l⊥m and m⊥n then l || n.
is perpendicular to
geometri
bottom element x = ⊥ means x is the smallest element. ∀x : x ∧ ⊥ = ⊥
the bottom element
lattice theory
|=
 entailment A ⊧ B means the sentence A entails the sentence B, that is every model in which A is true, B is also true. A ⊧ A ∨ ¬A
entails
model theory
|-
 inference x ⊢ y means y is derived from x. A → B ⊢ ¬B → ¬A
infers or is derived from
propositional logic, predicate logic
◅
 normal subgroup N ◅ G means that N is a normal subgroup of group G. Z(G) ◅ G
is a normal subgroup of
group theory
/
 quotient group G/H means the quotient of group G modulo its subgroup H. {0, a, 2a, b, b+a, b+2a} / {0, b} = {{0, b}, {a, b+a}, {2a, b+2a}}
mod
group theory
≈
 isomorphism G ≈ H means that group G is isomorphic to group H Q / {1, −1} ≈ V,
where Q is the quaternion group and V is the Klein four-group.
is isomorphic to
group theory
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