# Set Symbols

A set is a collection of things, usually numbers. We can list each element (or "member") of a set inside curly brackets like this: ## Common Symbols Used in Set Theory

Symbols save time and space when writing. Here are the most common set symbols

In the examples C = {1, 2, 3, 4} and D = {3, 4, 5}

Symbol Meaning Example
{ } Set: a collection of elements {1, 2, 3, 4}
A B Union: in A or B (or both) C D = {1, 2, 3, 4, 5}
A B Intersection: in both A and B C D = {3, 4}
A B Subset: every element of A is in B. {3, 4, 5} D
A B Proper Subset: every element of A is in B,
but B has more elements.
{3, 5} D
A B Not a Subset: A is not a subset of B {1, 6} C
A B Superset: A has same elements as B, or more {1, 2, 3} {1, 2, 3}
A B Proper Superset: A has B's elements and more {1, 2, 3, 4} {1, 2, 3}
A B Not a Superset: A is not a superset of B {1, 2, 6} {1, 9}
Ac Complement: elements not in A Dc = {1, 2, 6, 7}
When = {1, 2, 3, 4, 5, 6, 7}
A − B Difference: in A but not in B {1, 2, 3, 4} {3, 4} = {1, 2}
a A Element of: a is in A 3 {1, 2, 3, 4}
b A Not element of: b is not in A 6 {1, 2, 3, 4}
Empty set = {} {1, 2} {3, 4} = Ø Universal Set: set of all possible values
(in the area of interest)

P(A) Power Set: all subsets of A P({1, 2}) = { {}, {1}, {2}, {1, 2} }
A = B Equality: both sets have the same members {3, 4, 5} = {5, 3, 4}
A×B Cartesian Product
(set of ordered pairs from A and B)
{1, 2} × {3, 4}
= {(1, 3), (1, 4), (2, 3), (2, 4)}
|A| Cardinality: the number of elements of set A |{3, 4}| = 2

| Such that { n | n > 0 } = {1, 2, 3,...}
: Such that { n : n > 0 } = {1, 2, 3,...}
For All x>1, x2>x
There Exists x | x2>x
Therefore a=b b=a Natural Numbers {1, 2, 3,...} or {0, 1, 2, 3,...} Integers {..., −3, −2, −1, 0, 1, 2, 3, ...} Rational Numbers Algebraic Numbers Real Numbers Imaginary Numbers 3i Complex Numbers 2 + 5i