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}= ∅

- A ∪ A

- 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

- (A

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 )

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 }__Example__A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C )

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 }__Example__

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}

( A ∪ B )__Example:__^{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 }- x ∈ ( A ∪ B )

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}

( A ∩ B )__Example:__^{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 }- x ∈ ( A ∩ B )