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Daftar simbol matematika Simbol matematika dasar Nama Simbol Dibaca sebagai Penjelasan Contoh Kategori kesamaan = sama dengan x = y berarti x and y mewakili hal atau nilai yang sama. 1+1=2 x ≠ y berarti x dan y tidak mewakili hal atau nilai yang sama. 1≠2 umum Ketidaksamaan ≠ tidak sama dengan umum < ketidaksamaan lebih kecil dari; lebih besar dari x < y berarti x lebih kecil dari y. x > y means x lebih besar dari y. > order theory ≤ inequality ≥ 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. order theory 3<4 5>4 3 ≤ 4 and 5 ≤ 5 5 ≥ 4 and 5 ≥ 5 tambah tambah 4 + 6 berarti jumlah antara 4 dan 6. 2+7=9 aritmatika + disjoint union the disjoint union of A1 + A2 means the disjoint union of sets … and … A1 and A2. teori himpunan 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)} kurang kurang − 9 − 4 berarti 9 dikurangi 4. 8−3=5 −3 berarti negatif dari angka 3. −(−5) = 5 aritmatika tanda negatif negatif aritmatika set-theoretic complement minus; without A − B means the set that contains all the elements of A that are not in B. {1,2,4} − {1,3,4} = {2} set theory multiplication kali 3 × 4 means the multiplication of 3 by 4. 7 × 8 = 56 aritmatika Cartesian product × the Cartesian product of … and …; the direct product of … and … 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)} teori himpunan cross product cross u × v means the cross product of vectors (1,2,5) × (3,4,−1) = u and v (−22, 16, − 2) vector algebra ÷ division bagi 2 ÷ 4 = .5 6 ÷ 3 atau 6/3 berati 6 dibagi 3. / 12/4 = 3 aritmatika square root akar kuadrat √ √x berarti bilangan positif yang kuadratnya x. √4 = 2 bilangan real complex square root if z = r exp(iφ) is represented in polar the complex square coordinates with -π < φ ≤ π, then √z = √r √(-1) = i root of; square root exp(iφ/2). bilangan complex absolute value || absolute value of |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 n! is the product 1×2×...×n. 4! = 1 × 2 × 3 × 4 = 24 numbers ! factorial faktorial combinatorics ~ probability distribution has distribution X ~ D, means the random variable X has the probability distribution D. X ~ N(0,1), the standard normal distribution statistika ⇒ material implication A ⇒ B means if A is true then B is also true; if A is false then nothing is said implies; if .. then about B. → → may mean the same as ⇒, or it may have the meaning for functions given below. propositional logic ⊃ may mean the same as ⇒, or it may have the meaning for superset given below. ⊃ ⇔ material equivalence if and only if; iff ↔ ¬ ˜ A ⇔ B means A is true if B is true and A is false if B is false. logical negation not propositional logic 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. and logical disjunction or join in a lattice or propositional logic, lattice theory exclusive or xor ¬(¬A) ⇔ A x ≠ y ⇔ ¬(x = y) 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. 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 ∨ x + 5 = y +2 ⇔ x + 3 = y propositional logic logical conjunction or meet in a lattice ∧ x = 2 ⇒ x2 = 4 is true, but x2 = 4 ⇒ x = 2 is in general false (since x could be −2). The statement A ⊕ B is true when either (¬A) ⊕ A is always true, A or B, but not both, are true. A ⊻ B A ⊕ A is always false. ⊕ ⊻ means the same. propositional logic, Boolean algebra universal quantification ∀ for all; for any; for each ∀ x: P(x) means P(x) is true for all x. ∀ n ∈ N: n2 ≥ n. ∃ x: P(x) means there is at least one x such that P(x) is true. ∃ n ∈ N: n is even. ∃! x: P(x) means there is exactly one x such that P(x) is true. ∃! n ∈ N: n + 5 = 2n. 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). cosh x := (1/2)(exp x + exp (−x)) predicate logic ∃ existential quantification there exists predicate logic uniqueness quantification ∃! there exists exactly one predicate logic := definition is defined as ≡ everywhere P :⇔ Q means P is defined to be logically equivalent to Q. A XOR B :⇔ (A ∨ B) ∧ ¬(A ∧ B) :⇔ set brackets {,} the set of ... {a,b,c} means the set consisting of a, b, and c. N = {0,1,2,...} {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} ∅ berarti himpunan yang tidak memiliki elemen. {} juga berarti hal yang sama. {n ∈ N : 1 < n2 < 4} = ∅ teori himpunan {:} {|} set builder notation the set of ... such that ... teori himpunan himpunan kosong himpunan kosong ∅ teori himpunan {} ∈ set membership is an element of; is not an element of ∉ everywhere, teori himpunan ⊆ subset is a subset of ⊂ teori himpunan ⊇ superset ⊃ is a superset of teori himpunan a ∈ S means a is an element of the set S; a ∉ S means a is not an element of S. A ⊆ B means every element of A is also element of B. (1/2)−1 ∈ N 2−1 ∉ N A ∩ B ⊆ A; Q ⊂ R A ⊂ B means A ⊆ B but A ≠ B. A ⊇ B means every element of B is also element of A. A ∪ B ⊇ B; R ⊃ Q A ⊃ B means A ⊇ B but A ≠ B. set-theoretic union ∪ the union of ... and ...; 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 A ∩ B means the set that contains all those elements that A and B have in common. {x ∈ R : x2 = 1} ∩ N = {1} teori himpunan set-theoretic intersection ∩ intersected with; intersect teori himpunan \ set-theoretic complement minus; without A \ B means the set that contains all those {1,2,3,4} \ {3,4,5,6} = elements of A that are not in B. {1,2} teori himpunan function application () of teori himpunan f(x) berarti nilai fungsi f pada elemen x. Jika f(x) := x2, maka f(3) = 32 = 9. precedence grouping Perform the operations inside the parentheses first. (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4. 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. 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). N means {0,1,2,3,...}, but see the article on natural numbers for a different convention. {|a| : a ∈ Z} = N Z means {...,−3,−2,−1,0,1,2,3,...}. {a : |a| ∈ N} = Z umum arrow f:X→ function from ... to Y teori himpunan function composition o composed with teori himpunan natural numbers N N ℕ numbers integers Z Z ℤ numbers rational numbers Q Q ℚ numbers 3.14 ∈ Q Q means {p/q : p,q ∈ Z, q ≠ 0}. π∉Q real numbers R R ℝ numbers R means {limn→∞ an : ∀ n ∈ N: an ∈ Q, the limit exists}. π∈R √(−1) ∉ R complex numbers C C ℂ numbers C means {a + bi : a,b ∈ R}. i = √(−1) ∈ C infinity ∞ infinity numbers ∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits. limx→0 1/|x| = ∞ π berarti perbandingan (rasio) antara keliling lingkaran dengan diameternya. A = πr² adalah luas lingkaran dengan jari-jari (radius) r ||x|| is the norm of the element x of a normed vector space. ||x+y|| ≤ ||x|| + ||y|| pi π pi Euclidean geometry norm || || norm of; length of linear algebra summation ∑ sum over ... from ... ∑k=1n ak means a1 + a2 + ... + an. to ... of ∑k=14 k2 = 12 + 22 + 32 + 42 = 1 + 4 + 9 + 16 = 30 aritmatika product product 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 ∏i=0nYi means the set of all (n+1)-tuples (y0,...,yn). ∏n=13R = Rn aritmatika Cartesian product the Cartesian product of; the direct product of set theory derivative ' f '(x) is the derivative of the function f at … prime; derivative the point x, i.e., the slope of the tangent of … there. kalkulus If f(x) = x2, then f '(x) = 2x indefinite integral or antiderivative ∫ indefinite integral ∫ f(x) dx means a function whose of …; the derivative is f. antiderivative of … ∫x2 dx = x3/3 + C kalkulus definite integral ∫ab f(x) dx means the signed area between integral from ... to the x-axis and the graph of the function f ∫0b x2 dx = b3/3; ... of ... with respect between x = a and x = b. to kalkulus gradient ∇ del, nabla, gradient of ∇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) kalkulus partial derivative ∂ With f (x1, …, xn), ∂f/∂xi is the derivative If f(x,y) = x2y, then ∂f/∂x partial derivative of of f with respect to xi, with all other = 2xy variables kept constant. kalkulus boundary boundary of ∂M means the boundary of M ∂{x : ||x|| ≤ 2} = {x : || x || = 2} 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. topology perpendicular is perpendicular to ⊥ geometri bottom element the bottom element x = ⊥ means x is the smallest element. ∀x : x ∧ ⊥ = ⊥ lattice theory entailment |= entails model theory 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 x ⊢ y means y is derived from x. A → B ⊢ ¬B → ¬A inference |- infers or is derived from propositional logic, predicate logic normal subgroup ◅ is a normal subgroup of N ◅ G means that N is a normal subgroup Z(G) ◅ G of group G. group theory quotient group / mod group theory ≈ isomorphism is isomorphic to 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}} G ≈ H means that group G is isomorphic Q / {1, −1} ≈ V, to group H where Q is the quaternion group theory group and V is the Klein four-group.