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 | |||
≤
≥ | inequality | 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 | |||
+
| tambah | 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 | |||
−
| kurang | 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 | |||
| 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. |
| or | |||
| 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 fbetween 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 |
