# Lesson 1: Set Theory

## Introduction to Sets

A set is a model for an unordered collection of things in mathematics, such as integer numbers or symbols. A set can be empty, have a single element, or have infinite elements.

This lesson introduces the Notation and terminology of set theory which is basic and used throughout this course.

## Subsets, Sets, and Elements

A set is a model of an unordered collection of things. For Example:

Students in a classroom belong to a set.

Every object in a set is called an element. For Example, a student is an element from the students’ set.

The notation x ∈ A denotes that x is an element of Set A.

The notation x ∉ A denotes that x is not an element of Set A.

## Notations to Describe a Set

In mathematics, there are three Notations to describe a set:

### 1- Roster Method

In the roster method notation, the elements of a set are listed inside brackets. Examples:

S = {a,d,c,b}

S = {a,b,c,d,e, f} = {a,b,c,b,c,d,e,f} , no repeated element

S = {a,b,c,d, e,f … ,z }, series

S = {{a,b,c}, r, {e,f}} , sets within a set

### 2- Builder Method

In the builder methods, the Set is described by representing the elements or explaining the content of the Set. For Example:

S = {x | x is a positive integer less than 50}, is a set of integers between 0 and 49

S = {x | x >= 5 or x < 0}, a set of numbers between 1 and 5

### 3- Interval Notation

The third set notation type simply represents an interval of numbers. For Example:

[2,6] = {x | 2 ≤ x ≤ 6} , close interval, means the endpoint 2 and 6 are not included.

([2,6) = {x | 2 ≤ x ≤ 6} , openinterval, means the endpoint 2 and 6 are included.

## Universal Set and Empty Set

Unless explicitly or implicitly said otherwise, all sets are assumed to be members of a fixed huge set called the universal Set, which we denote by U.

An empty Set is simply a set with no elements symbolized . For Example:

s = ∅

A set contains an empty set ∅, is not empty. For Example:

S = { ∅ } is not an empty set

## Venn Diagram

A Venn diagram shows the relation between a collection of sets. For example, if we have two sets, ‘A’ and ‘B’, the intersection of ‘A’ and ‘B’ is represented as the following:

## Common Universal Sets

Some of the commonly used sets in mathematics:

Integers set

Denoted by Z. For example:  Z = {…,-5,-4,-3,-2,-1,0,1,2,3,4,5…}

Natural numbers set

Donated by N. For Example:  N = {0,1,2,3,4,5….}

Positive integers set

Donated by Z⁺. For example: Z⁺ = {1,2,3,4,5…..}

Set of real numbers

Donated by R. For example: R = Set of real numbers

Set of positive real numbers

Donated by R+. For example: R+ = set of positive real numbers

Set of complex numbers

Donated by C. For Example: C = set of complex numbers.

Set of rational numbers

Donated by Q. For example: Q = Set of rational numbers

## Set Examples

The sets below represent even integers using roster, builder, and interval notations.

Roster: {…,-4,-2,0,2,4,….}

Builder: {x | x=2n, n ∊ Z}

Interval: [a,b] = {x | a ≤ x ≤ b}

## Set Equality and Subsets

Two sets are equal if and only if they have the same elements.

A = B if  \forall x (x ∊ A\leftrightarrow x ∊ B) (\forall stands for “For all”)

Examples: {1,2,3} = {3, 5, 1} , {1,2,3,3,2,1} = {1,2,3}

The set A is a subset of B, if and only if every element of A is also an element of B.

• A = B if and only if \forall x (x ∊ A\rightarrow x ∊ B)
• For any set A, ∅ ⊆ A, and A ⊆ A.
• If A ⊆ B and B ⊆  A, then A = B.

Venn diagram

## Proper Subsets and Cardinality

### Proper subset

If a set B is a subset of A, donated by A ⊆ B, and A is not equal to B, then A is a proper subset of B, A ⊂ B. For Example: if A = {1, 3,5} and B ={1,3}, B is a proper subset of A.

### Cardinality

The cardinality of a set A, denoted by |A|, is the number of unique elements of A. Examples: |{1,2,3}| = 3, The set N is infinite.

Note
The set must be finite to get its cardinality.

## Element V.S. Subset

Sets can be elements or subsets of other sets.

For example, if A = {4,6,{4},{8}}, then

4 is an element of A, while {4} is a set of A.

The below representation are true:

• 4 ∈ A ( Element 4 belongs to Set A)
• {4} ∈ A ( Set {4} belongs to set of A)
• {4} ⊂ A ( 4 is a proper subset of A )
• {{4}} ⊂ A ({4} is a proper subset of A)
• 6 ∈ A (Element 6 belongs to Set A)
• {6} ∉ A (Set {6} does not belong to Set A)
• {6} ⊂ A ( 6 is proper subset of A)
• {{6}} ⊄ A (Set {6} is note a proper subset of A)
• 8 ∉ A (Element 8 does not belong to Set A)
• {8} ∈ A (Set {8} belongs to Set A)
• {8} ⊄ A (Set 6 is not a proper subset of A)
• {{8}} ⊂ A (Set {8} is a proper subset of A)

Note: ⊂ is the symbol for a proper subset, ⊄ is for ‘not a subset’

## The Power Set

A power set of the set A, donated by P(A), is the set of all subsets of A

Example: If A = {a,b} then

P(A) = {ø, {a},{b},{a,b}}

The total number of subsets = | P(A) | = 2|A| (power a)

Example, the number of subsets P(A) if A ={a,b) is 2 power 2 = 4.

## Cartesian Product

The Cartesian product of two sets A and B (A x B), is the set of all possible ordered pairs (a,b), where a ∈ A and b ∈ B:

For example:

If A = {1,2,3} and B = {a,b}

The cartesian product:

A x B = {(1,a),(2,a),(3,a),(1,b),(2,b),(3,b)}

B x A = {(a,1), (b,1),(a,2).(b,2),(a,3),(b,3)}

## Quick revision

Consider the following sets:

A = {a,b,c} , B={1,2}, and C={5}, then:

• A X B = {(a,1),(a,2), (b,1),(b,2),(c,1),(c,2)}
• B X C = {(1,5, (2,5)}
• C X B = {(5,1), (5,2)}
• |A X B| = 6
• {b,1} ∉ A X B
• (c,2) ∈ A X B
• B X C ≠ C X B

sets

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