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:

Set Notation

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: A has some (or all) elements of B {3,4,5} D
A B Proper Subset: A has some elements of B {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 set universal = {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
set universal 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} = {3,4,5}
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
     
set natural Natural Numbers {1,2,3,...} or {0,1,2,3,...}
set integer Integers {..., -3, -2, -1, 0, 1, 2, 3, ...}
set rational Rational Numbers  
set algebraic Algebraic Numbers  
set real Real Numbers  
set imaginary Imaginary Numbers 3i
set complex Complex Numbers 2 + 5i