| Symbol | Name | Explanation | Example |
| Should be read as |
| Category |
| = | equality | x = y means x and y represent the same thing or value. | 1 + 1 = 2 |
| is equal to; equals |
| everywhere |
| ≠ | Inequation | x ≠ y means that x and y do not represent the same thing or value. | 1 ≠ 2 |
| is not equal to; does not equal |
| everywhere |
| + | addition | 4 + 6 means the sum of 4 and 6. | 2 + 7 = 9 |
| plus |
| arithmetic |
| − | subtraction | 9 − 4 means the subtraction of 4 from 9. | 8 − 3 = 5 |
| minus |
| arithmetic |
| negative sign | −3 means the negative of the number 3. | −(−5) = 5 |
| negative |
| arithmetic |
| set theoretic complement | A − B means the set that contains all the elements of A that are not in B. | {1,2,4} − {1,3,4} = {2} |
| minus; without |
| set theory |
| × | multiplication | 3 × 4 means the multiplication of 3 by 4. | 7 × 8 = 56 |
| times |
| arithmetic |
| 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 … |
| set theory |
| ÷
/ | division | 6 ÷ 3 or 6/3 means the division of 6 by 3. | 2 ÷ 4 = .5
12/4 = 3 |
| divided by |
| arithmetic |
| ⇒
→
⊃ | 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 | is true when either A or B is true, but not when both are true. | (¬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 + 5 = 2n |
| there exists |
| 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 ... |
| set theory |
| { : }
{ | } | 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 ... |
| set theory |
∅
{} | empty set | ∅ means the set with no elements. {} means the same. | {n ∈ N : 1 < n2 < 4} = ∅ |
| the empty set |
| set theory |
| ∈
∉ | 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, set theory |
| ⊆
⊂ | 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 |
| set theory |
| ⊇
⊃ | 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 |
| set theory |
| ∪ | 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 |
| set theory |
| ∩ | 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 |
| set theory |
| \ | 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 |
| set theory |
| ( ) | function application | f(x) means the value of the function f at the element x. | If f(x) := x2, then f(3) = 32 = 9. |
| of |
| set theory |
| precedence grouping | Perform the operations inside the parentheses first. | (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4. |
|
| everywhere |
| 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 |
| functions |
N
ℕ | natural numbers | N means {0,1,2,3,...}, but see the article on natural numbers for a different convention. | {|a| : a ∈ Z} = N |
| N |
| numbers |
Z
ℤ | integers | Z means {...,−3,−2,−1,0,1,2,3,...}. | {a : |a| ∈ N} = Z |
| Z |
| numbers |
Q
ℚ | rational numbers | Q means {p/q : p,q ∈ Z, q ≠ 0}. | 3.14 ∈ Q
π ∉ Q |
| Q |
| numbers |
R
ℝ | real numbers | R means {limn→∞ an : ∀ n ∈ N: an ∈ Q, the limit exists}. | π ∈ R
√(−1) ∉ R |
| R |
| numbers |
C
ℂ | complex numbers | C means {a + bi : a,b ∈ R}. | i = √(−1) ∈ C |
| C |
| numbers |
| <
> | strict inequality | x < y means x is less than y.
x > y means x is greater than y. | x < y ⇔ y > x |
| is less than, is greater than |
| partial orders |
| ≤
≥ | inequality | x ≤ y means x is less than or equal to y.
x ≥ y means x is greater than or equal to y. | x ≥ 1 ⇒ x2 ≥ x |
| is less than or equal to, is greater than or equal to |
| partial orders |
| √ | square root | √x means the positive number whose square is x. | √(x2) = |x| |
| the principal square root of; square root |
| real numbers |
| ∞ | 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 means the ratio of a circle's circumference to its diameter. | A = πr² is the area of a circle with radius r |
| pi |
| Euclidean geometry |
| ! | factorial | n! is the product 1×2×...×n. | 4! = 24 |
| factorial |
| combinatorics |
| | | | absolute value | |x| means the distance in the real line (or the complex plane) between x and zero. | |a + bi| = √(a2 + b2) |
| absolute value of |
| numbers |
| || || | norm | ||x|| is the norm of the element x of a normed vector space. | ||x+y|| ≤ ||x|| + ||y|| |
| norm of; length of |
| functional analysis |
| ∑ | 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 |
| arithmetic |
| ∏ | 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 |
| arithmetic |
| 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 |
| ∫ | integration | ∫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; ∫x2 dx = x3/3 |
| integral from ... to ... of ... with respect to |
| calculus |
| f ' | 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 and f ''(x) = 2 |
| derivative of f; f prime |
| calculus |
| ∇ | 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)
A transparent image for text is: Image:Del.gif ( ). |
| del, nabla, gradient of |
| calculus |
| ∂ | 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 |
| calculus |
| ⊥ | perpendicular | x ⊥ y means x is perpendicular to y; or more generally x is orthogonal to y. | |
| is perpendicular to |
| orthogonality |
| bottom element | x = ⊥ means x is the smallest element. | |
| the bottom element |
| lattice theory |
| ⊧ | entailment | means the sentence a entails the sentence b. Formal definition: if and only if, in every model in which a is true, b is also true. | |
| entails |
| propositional logic, predicate logic |
| ⊢ | inference | x y means y is derived from x. | |
| infers or is derived from |
| propositional logic, predicate logic |
The article wikipedia: How does one edit a page contains information about how to produce these math symbols in Wikipedia articles.
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