Be aware that, in some cases, different symbols have the same meaning, and the same symbol has, depending on the context, different meanings.
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
| {1,2,3,4} − {3,4,5,6} = {1,2}
|
| 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
| <math>A \oplus B<math> is true when either A or B are true, but not when both are true
| (¬A) <math> \oplus <math> A is always true, A <math> \oplus <math> 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 theory|set theory]]
| {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 ...
|
| [[naive set
|
∅
{} | empty set
| {} means the set with no elements; ∅ is the same thing
| {n ∈ N : 1 < n2 < 4} = {}
|
| 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
| Consider the function f: Z → N 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
| 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
| <math>a \models b<math> means the sentence a entails the sentence b. Formal definition: <math>a \models b<math> if and only if, in every model in which a is true, b is also true.
|
|
| entails
|
| propositional logic, predicate logic
|
| ⊢
| inference
| x<math> \vdash <math> y means y is derived from x.
|
|
| infers or is derived from
|
| propositional logic, predicate logic
|
