# Discrete Math 1: Set Theory

# 1. Definitions

// Set A contains elements 1,2 and 3

A = {1,2,3}// 2 is an element of A

2∈A// 4 is not an element of A

4∉A

# 2. Number Sets

`Naturals N = {1,2,3,4,...}`

Integers Z = {...,-2,-1,0,1,2,...}

Rationals Q = Ratio of 2 integers. example: 1/2

Irrationals Q′= Can't be represented as ratios of integers

Real R = Q and Q′

Imaginary I = Everything not in the reals. Ex: (√x = -1)

Complex C = Reals and imaginaries

# 3. Set Equality

The order and repetition of elements does not matter

A = {1,2,3}

B = {2,3,2,2,3,1,2,3,2,1,1,2,3,2,2,3,1,1,3,2,1}A = B

# 4. Set Builder Notation

A = {1,2,3,4,5,6,7,8,9}// B is equal to all elements of A (x∈A), such that (|),

// the elements are less than 5 (x<5)

B = { x∈A | x<5 }B = {1,2,3,4}

# 5. Types of Sets

`Universal U `

Empty {} or ϕ

Finite

Infinite

Subset

# 6. Cardinality

The number of distinct elements in a set.

`n(A) or |A|`

# 7. Equivalence

Sets are equivalent when their cardinality is the same. **NOT to be mistaken with equality.**

A = {1,2,3,4}

B = {5,6,7,8}// A and B are equivalent

A~B

# 8. Subsets

Subsets: ⊆

**Proper** subsets: ⊂

A = {1,2,3,4}

B = {2,3}// A is a subset of A, but not a PROPER subset of A

// ⊆

A⊆A// B is a proper subset of AB⊂A

# 9. Power Sets

A power set P(A) is the set of all of the subsets of A.

`A = {1,2,3}`

P(A) = {{},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}