Simbol matematika dasar

Dalam matematika sering digunakan simbol-simbol yang umum dikenal oleh matematikawan. Sering kali pengertian simbol ini tidak dijelaskan, karena dianggap maknanya telah diketahui. Hal ini kadang menyulitkan bagi mereka yang awam. Daftar berikut ini berisi banyak simbol beserta artinya.

Simbol matematika dasar

Simbol	Nama	Penjelasan	Contoh
	Dibaca sebagai
	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

dikutip dari
http://id.wikipedia.org/wiki/Daftar_simbol_matematika