This is a work in progress. Its intended to be used as a quick reference.
Set Notation
∅ = empty set
 ∅ = {}
N = natural numbers

N = { 1, 2, 3, … }

N_{0} = { 0, 1, 2, 3, … }
Z = integers
 Z = { …, 1, 0, 1, … }
Q = rational numbers
R = real numbers
C = complex numbers
Set Operators
∪ = union

A ∪ B = { p : p ∈ A or p ∈ B }
 The definition of A union B equals the set containing elements p such that p is an element of A or p is an element of B.
∩ = intersection

A ∩ B = { p : p ∈ A and p ∈ B }
 The definition of A intersection B equals the set containing elements p such that p is an element of A and p is an element of B.
\ = complement

A \ B = { p : p ∈ A or p ∉ B }
 The definition of A difference of B equals the set containing elements p such that p is an element of A or p is not an element of B.
 xor
 A ⊕ B = ( A \ B ) ∪ ( B \ A )
R \ Q = I
(Some) Laws of Set Theory
U = the universe

Applying a binary operator to a proposition over and over will never change the value of the original proposition.
 A ∪ A = A
 A ∩ A = A

An equality relation, A and B produce the same value as each other.
 A ∪ ∅ = A
 U ∪ A = U
 A ∩ U = A
 ∅ ∩ A = ∅

A^{c} = { x : x ∉ A }
 A ∪ A^{c} = U
 ∅^{c} = U
 A ∩ A^{c} = ∅
 U^{c} = ∅

If we have a proposition and apply it to the same function twice it will yield the original value.

A function that is its own inverse.
 (A^{c})^{c} = A
 f(f(x)) = x

Sets can be regrouped.
 A ∩ ( B ∩ C ) = ( A ∩ B) ∩ C
 A ∪ ( B ∪ C ) = ( A ∪ B) ∪ C

The sets can be switched relative to the operator.
 A ∩ B = B ∩ A
 A ∪ B = B ∪ A

A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C )

To show equality, you must prove that each side of an equation are subsets of each other:
 A ∩ ( B ∪ C ) ⊆ ( A ∩ B ) ∪ ( A ∩ C )
 ( A ∩ B ) ∪ ( A ∩ C ) ⊆ A ∩ ( B ∪ C )
Example A = { 1, x, 3 } B = { 3, k, z } C = { 1, 3 } A ∩ B = { 3 } < A set with only one element is referred A ∩ C = { 1, 3 } to as a singleton or a singleton set. B ∪ C = { 1, 3, k, z } A ∩ ( B ∪ C ) = { 1, 3 } A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) { 1, 3 } = { 3 } ∪ { 1, 3} { 1, 3 } = { 1, 3 }


A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C )
Example A = { 1, x, 3 } B = { 3, k, z } C = { 1, 3 } A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C ) A ∪ B = { 1, x, 3, k, z } A ∪ C = { 1, x, x } B ∩ C = { 3 } A ∪ ( B ∩ C ) = { 1, x, 3 } A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C ) { 1, x, 3 } = { 1, x, 3, k, z } ∩ { 1, x, 3 } { 1, x, 3 } = { 1, x, 3 }
The complement of the union is the intersection of the complements.

( A ∪ B )^{c} ⊆ A^{c} ∩ B^{c}
 x ∈ ( A ∪ B )^{c}
 x ∉ ( A ∪ B )
 x ∉ A and x ∉ B
 x ∈ A^{c} and x ∈ B^{c}
 x ∈ A^{c} ∩ B^{c}
Example: ( A ∪ B )^{c} ⊆ A^{c} ∩ B^{c} A = { 1, x, 3 } B = { 3, k, z } C = { 1, 3 } U = { x ∈ Z : 0 ≤ x ≤ 5 } A ∪ B = { 1, x, 3, k, z } ( A ∪ B )^{c} = { 0, 2, 4, 5 } A^{c} ∩ B^{c} = { 0, 2, 4, 5 }
The complement of the intersection is the union of the complements.

( A ∩ B )^{c} ⊆ A^{c} ∪ B^{c}
 x ∈ ( A ∩ B )^{c}
 x ∉ ( A ∩ B )
 x ∉ A or x ∉ B
 x ∈ A^{c} or x ∈ B^{c}
 x ∈ A^{c} ∪ B^{c}
Example: ( A ∩ B )^{c} ⊆ A^{c} ∪ B^{c} A = { 1, x, 3 } B = { 3, k, z } C = { 1, 3 } U = { x ∈ Z : 0 ≤ x ≤ 5 } A ∩ B = { 3 } ( A ∩ B )^{c} = { 0, 1, 2, 4, 5 } A^{c} ∪ B^{c} = { 0, 1, 2, 4, 5 }