Symbol 
Name 
reads as 
Category 
+
 addition
 plus
 arithmetic

4 + 6 = 10 means that if four is added to 6, the sum, or result, is 10. 
43 + 65 = 108; 2 + 7 = 9


 subtraction
 minus
 arithmetic

9  4 = 5 means that if 4 is subtracted from 9, the result will be 5. The  sign is unique in that it can also denote that a number is negative. For example, 5 + (3) = 2 means that if five and negative three are added, the result is two. 
87  36 = 51

⇒ →
 material implication
 implies; if .. then
 propositional logic

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 mentioned further down

x = 2 ⇒ x^{2} = 4 is true, but x^{2} = 4 ⇒ x = 2 is in general false (since x could be −2)

⇔ ↔
 material equivalence
 if and only if; iff
 propositional logic

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

∧
 logical conjunction
 and
 propositional logic

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

∨
 logical disjunction
 or
 propositional logic

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

¬ /
 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

¬(A ∧ B) ⇔ (¬A) ∨ (¬B); x ∉ S ⇔ ¬(x ∈ S)

∀
 universal quantification
 for all; for any; for each
 predicate logic

∀ x: P(x) means: P(x) is true for all x 
∀ n ∈ N: n^{2} ≥ n 
∃
 existential quantification
 there exists
 predicate logic

∃ x: P(x) means: there is at least one x such that P(x) is true

∃ n ∈ N: n + 5 = 2n

=
 equality
 equals
 everywhere

x = y means: x and y are different names for precisely the same thing

1 + 2 = 6 − 3

:= :⇔
 definition
 is defined as
 everywhere

x := y means: x is defined to be another name for y 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)

{ , }
 set brackets
 the set of ...
 set theory

{a,b,c} means: the set consisting of a, b, and c

N = {0,1,2,...}

{ : } {  }
 set builder notation
 the set of ... such that ...
 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 : n^{2} < 20} = {0,1,2,3,4}

∅ {}
 empty set
 empty set
 set theory

{} means: the set with no elements; ∅ is the same thing

{n ∈ N : 1 < n^{2} < 4} = {}

∈ ∉
 set membership
 in; is in; is an element of; is a member of; belongs to
 set theory

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

⊆ ⊂
 subset
 is a subset of
 set theory

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

∪
 set theoretic union
 the union of ... and ...; union
 set theory

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

∩
 set theoretic intersection
 intersected with; intersect
 set theory

A ∩ B means: the set that contains all those elements that A and B have in common

{x ∈ R : x^{2} = 1} ∩ N = {1}

\\
 set theoretic complement
 minus; without
 set theory

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}

( ) [ ] { }
 function application; grouping
 of
 set theory

for function application: f(x) means: the value of the function f at the element x for grouping: perform the operations inside the parentheses first

If f(x) := x^{2}, then f(3) = 3^{2} = 9; (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4

f:X→Y
 function arrow
 from ... to
 functions

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) = x^{2}

N
 natural numbers
 N
 numbers

N means: {0,1,2,3,...}

{a : a ∈ Z} = N

Z
 integers
 Z
 numbers

Z means: {...,−3,−2,−1,0,1,2,3,...}

{a : a ∈ N} = Z

Q
 rational numbers
 Q
 numbers

Q means: {p/q : p,q ∈ Z, q ≠ 0}

3.14 ∈ Q; π ∉ Q

R
 real numbers
 R
 numbers

R means: {lim_{n→∞} a_{n} : ∀ n ∈ N: a_{n} ∈ Q, the limit exists}

π ∈ R; √(−1) ∉ R

C
 complex numbers
 C
 numbers

C means: {a + bi : a,b ∈ R}

i = √(−1) ∈ C

< >
 comparison
 is less than, is greater than
 partial orders

x < y means: x is less than y; x > y means: x is greater than y

x < y ⇔ y > x

≤ ≥
 comparison
 is less than or equal to, is greater than or equal to
 partial orders

x ≤ y means: x is less than or equal to y; x ≥ y means: x is greater than or equal to y

x ≥ 1 ⇒ x^{2} ≥ x

√
 square root
 the principal square root of; square root
 real numbers

√x means: the positive number whose square is x

√(x^{2}) = x

∞
 infinity
 infinity
 numbers

∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits

lim_{x→0} 1/x = ∞

π
 pi
 pi
 Euclidean geometry

π means: the ratio of a circle's circumference to its diameter

A = πr² is the area of a circle with radius r

!
 factorial
 factorial
 combinatorics

n! is the product 1×2×...×n

4! = 12

 
 absolute value
 absolute value of
 numbers

x means: the distance in the real line (or the complex plane) between x and zero

a + bi = √(a^{2} + b^{2})

 
 norm
 norm of; length of
 functional analysis

x is the norm of the element x of a normed vector space

x+y ≤ x + y

∑
 summation
 sum over ... from ... to ... of
 arithmetic

∑_{k=1}^{n} a_{k} means: a_{1} + a_{2} + ... + a_{n}

∑_{k=1}^{4} k^{2} = 1^{2} + 2^{2} + 3^{2} + 4^{2} = 1 + 4 + 9 + 16 = 30

∏
 product
 product over ... from ... to ... of
 arithmetic

∏_{k=1}^{n} a_{k} means: a_{1}a_{2}···a_{n}

∏_{k=1}^{4} (k + 2) = (1 + 2)(2 + 2)(3 + 2)(4 + 2) = 3 × 4 × 5 × 6 = 360

∫
 integration
 integral from ... to ... of ... with respect to
 calculus

∫_{a}^{b} f(x) dx means: the signed area between the xaxis and the graph of the function f between x = a and x = b

∫_{0}^{b} x^{2} dx = b^{3}/3; ∫x^{2} dx = x^{3}/3

f '
 derivative
 derivative of f; f prime
 calculus

f '(x) is the derivative of the function f at the point x, i.e. the slope of the tangent there

If f(x) = x^{2}, then f '(x) = 2x

∇
 gradient
 del, nabla, gradient of
 calculus

∇f (x_{1}, …, x_{n}) is the vector of partial derivatives (df / dx_{1}, …, df / dx_{n})

If f (x,y,z) = 3xy + z² then ∇f = (3y, 3x, 2z)
A transparent image for text is: Image:Del.gif ().

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