Table of mathematical symbols

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

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

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 ⇒ x² = 4 is true, but x² = 4 ⇒ x = 2 is in general false (since x could be −2)

⇔
↔

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

x + 5 = y + 2 ⇔ x + 3 = y

∧

logical conjunction

and

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

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

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: n² ≥ 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 ...

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

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

{ : }
{ | }

set builder notation

the set of ... such that ...

{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}

∅
{}

empty set

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

{n ∈ N : 1 < n² < 4} = {}

∈
∉

set membership

in; is in; is an element of; is a member of; belongs to

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

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

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

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

{x ∈ R : x² = 1} ∩ N = {1}

\

set theoretic complement

minus; without

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

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) := x², then f(3) = 3² = 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) = x²

N

natural numbers

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

{|a| : a ∈ Z} = N

Z

integers

Z means: {...,−3,−2,−1,0,1,2,3,...}

{a : |a| ∈ N} = Z

Q

rational numbers

Q means: {p/q : p,q ∈ Z, q ≠ 0}

3.14 ∈ Q; π ∉ Q

R

real numbers

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

π ∈ R; √(−1) ∉ R

C

complex 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 ⇒ x² ≥ x

√

square root

the principal square root of; square root

real numbers

√x means: the positive number whose square is x

√(x²) = |x|

∞

infinity

∞ 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

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

combinatorics

n! is the product 1×2×...×n

4! = 12

| |

absolute value

absolute value of

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

|a + bi| = √(a² + b²)

|| ||

norm

norm of; length of

functional analysis

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

||x+y|| ≤ ||x|| + ||y||

∑

addition

sum over ... from ... to ... of

∑_k=1ⁿ a_k means: a₁ + a₂ + ... + a_n

∑_k=1⁴ k² = 1² + 2² + 3² + 4² = 1 + 4 + 9 + 16 = 30

∏

multiplication

product over ... from ... to ... of