Simbol | Nama | Penjelasan | Contoh | |
Dibaca sebagai | ||||
Kategori | ||||
= | x = y berarti x and y mewakili hal atau nilai yang sama. | 1 + 1 = 2 | ||
sama dengan | ||||
umum | ||||
≠ | x ≠ y berarti x dan y tidak mewakili hal atau nilai yang sama. | 1 ≠ 2 | ||
tidak sama dengan | ||||
umum | ||||
< > | 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 | ||||
≤ ≥ | 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 | ||||
+ | 4 + 6 berarti jumlah antara 4 dan 6. | 2 + 7 = 9 | ||
tambah | ||||
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 … | ||||
− | 9 − 4 berarti 9 dikurangi 4. | 8 − 3 = 5 | ||
kurang | ||||
−3 berarti negatif dari angka 3. | −(−5) = 5 | |||
negatif | ||||
A − B berarti himpunan yang mempunyai semua anggota dari A yang tidak terdapat pada B. | {1,2,4} − {1,3,4} = {2} | |||
minus; without | ||||
× | 3 × 4 berarti perkalian 3 oleh 4. | 7 × 8 = 56 | ||
kali | ||||
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 … | ||||
u × v means the cross product of vectors u and v | (1,2,5) × (3,4,−1) = (−22, 16, − 2) | |||
cross | ||||
÷ / | 6 ÷ 3 atau 6/3 berati 6 dibagi 3. | 2 ÷ 4 = .5 12/4 = 3 | ||
bagi | ||||
√ | √x berarti bilangan positif yang kuadratnya x. | √4 = 2 | ||
akar kuadrat | ||||
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 | ||||
| | | |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 | ||||
! | n! adalah hasil dari 1×2×...×n. | 4! = 1 × 2 × 3 × 4 = 24 | ||
faktorial | ||||
~ | X ~ D, means the random variable X has the probability distribution D. | X ~ N(0,1), the standard normal distribution | ||
has distribution; tidk terhingga | ||||
⇒ → ⊃ | x = 2 ⇒ x2 = 4 is true, but x2 = 4 ⇒ x = 2 is in general false (since x could be −2). | |||
implies; if .. then | ||||
⇔ ↔ | 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 | ||||
¬ ˜ | 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 | ||||
∧ | logical conjunction or meet in a lattice | The statement A ∧ B is true if A and B are both true; else it is false. | ||
and | ||||
∨ | 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. | \ | |
⊕ ⊻ | 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 | ||||
∀ | ∀ x: P(x) means P(x) is true for all x. | ∀ n ∈ N: n2 ≥ n. | ||
for all; for any; for each | ||||
∃ | ∃ x: P(x) means there is at least one x such that P(x) is true. | ∃ n ∈ N: n is even. | ||
there exists | ||||
∃! | ∃! x: P(x) means there is exactly one x such that P(x) is true. | ∃! n ∈ N: n + 5 = 2n. | ||
there exists exactly one | ||||
:= ≡ :⇔ | 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 ... | ||||
{ : } { | } | {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 ... | ||||
∅ {} | ∅ berarti himpunan yang tidak memiliki elemen. {} juga berarti hal yang sama. | {n ∈ N : 1 < n2 < 4} = ∅ | ||
himpunan kosong | ||||
∈ ∉ | 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 | ||||
⊆ ⊂ | 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 | ||||
⊇ ⊃ | 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 | ||||
∪ | 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 | ||||
∩ | 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 | ||||
\ | 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 | ||||
( ) | function application | f(x) berarti nilai fungsi f pada elemen x. | Jika f(x) := x2, maka f(3) = 32 = 9. | |
of | ||||
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 | ||||
o | 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 | ||||
N ℕ | N berarti {0,1,2,3,...}, but see the article on natural numbers for a different convention. | {|a| : a ∈ Z} = N | ||
N | ||||
Z ℤ | Z berarti {...,−3,−2,−1,0,1,2,3,...}. | {a : |a| ∈ N} = Z | ||
Z | ||||
Q ℚ | Q berarti {p/q : p,q ∈ Z, q ≠ 0}. | 3.14 ∈ Q π ∉ Q | ||
Q | ||||
R ℝ | R berarti {limn→∞ an : ∀ n ∈ N: an ∈ Q, the limit exists}. | π ∈ R √(−1) ∉ R | ||
R | ||||
C ℂ | C means {a + bi : a,b ∈ R}. | i = √(−1) ∈ C | ||
C | ||||
∞ | ∞ 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 | ||||
π | π berarti perbandingan (rasio) antara keliling lingkaran dengan diameternya. | A = πr² adalah luas lingkaran dengan jari-jari (radius) r | ||
pi | ||||
|| || | ||x|| is the norm of the element x of a normed vector space. | ||x+y|| ≤ ||x|| + ||y|| | ||
norm of; length of | ||||
∑ | ∑k=1n ak means a1 + a2 + ... + an. | ∑k=14 k2 = 12 + 22 + 32 + 42 = 1 + 4 + 9 + 16 = 30 | ||
sum over ... from ... to ... of | ||||
∏ | ∏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 | ||||
∏i=0nYi means the set of all (n+1)-tuples (y0,...,yn). | ∏n=13R = Rn | |||
the Cartesian product of; the direct product of | ||||
' | If f(x) = x2, then f '(x) = 2x | |||
… prime; derivative of … | ||||
∫ | ∫ f(x) dx means a function whose derivative is f. | ∫x2 dx = x3/3 + C | ||
indefinite integral of …; the antiderivative of … | ||||
∫0b x2 dx = b3/3; | ||||
integral from ... to ... of ... with respect to | ||||
∇ | ∇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) | ||
∂ | 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 | ||||
∂M means the boundary of M | ∂{x : ||x|| ≤ 2} = {x : || x || = 2} | |||
boundary of | ||||
⊥ | 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 | ||||
x = ⊥ means x is the smallest element. | ∀x : x ∧ ⊥ = ⊥ | |||
the bottom element | ||||
|= | 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 | ||||
|- | x ⊢ y means y is derived from x. | A → B ⊢ ¬B → ¬A | ||
infers or is derived from | ||||
◅ | N ◅ G means that N is a normal subgroup of group G. | Z(G) ◅ G | ||
is a normal subgroup of | ||||
/ | 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 | ||||
≈ | 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 | ||||
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