###### Set

A **set** is a collection of well-defined objects

A set is **well-defined** if we are able to tell whether or not any particular object is an element of the set

**Example:**

Which of the following are well-defined sets?

(1) All the tall girls of the school

(2) All the letters in the word **"mathematics"**

(3) All the hardworking teachers in a school

(4) All the honest members in the family

**Answer: (2)**

All the letters in the word **"mathematics"** is well-defined sets

B = {m, a, t, h, e, i, c, s}

The following conventions are used with sets:

- Capital letters are used to denote sets
- Curly braces { } denote a list of elements in a set
- Lowercase letters are used to denote elements of sets

**Example:**

A is the set of vowels

A = {a, i, u, e, o}

The elements in the sets are depicted in either the statement form, set-builder notation form or roster form

1. Statement form

A = {prime numbers between 10 and 30}

2. Set-builder notation

A = {x| 10 < x < 30, x ∈ prime numbers}

We read it as, "A is the set of all x such that x is more than10 but less than 30 and x belongs to prime numbers"

3. Roster form

A = {11, 13, 17, 19, 23, 29}

##### Elements Of a Set

An element is a member of a set.

Notation:

∈ means **“is an element of”**

∉ means **“is not element of”**

##### Cardinality

The number of elements of a set is expressed as n(A)

Example:

A = {positive integers between −3 and 3}

A = {1, 2}

1 ∈ A {1 is an element of A}

2 ∈ A {2 is an element of A}

3 ∉ A {3 is not element of A}

n(A) = 2

##### Empty Set and Infinite Set

**Null or empty set **

Set with no elements

Example:

The sets of triangles having 4 sides

It is an empty set, because every triangle has 3 sides

**Infinite set**

The element cannot be listed (unlimited)

Example:

- Set of all point in a line segment
- Set of all natural numbers
- Set of all integers

##### Subset

\(\text{A}\) is a subset of \(\text{A}\) (\(\text{A}\subseteq \text{B}\)) if and only if every element of \(\text{A}\) is in \(\text{B}\).

The total number of subsets of a set with n elements is \(2^{n}\).

**Example:**

\(\text{P} = \lbrace 5, 10, 15, 20 \rbrace\)

\(\text{Q} = \lbrace 5, 10, 15, 20, 25, 30 \rbrace\)

\(\text{P}\) is a** proper subset** of \(\text{Q}\) because the element 25 and 30 is not in the set \(\text{P}\). (\(\text{P} \subset \text{Q}\))

The number of subset of \(\text{Q}\) is \(2^6 = 64\).

**Example:**

Find the number of subsets of \(\text{P} = \lbrace \text{a, b, c, d, e, f}\rbrace\) that contain exactly 3 elements

The total number of subsets of \(\text{P}\) is \(2^{6}\)

Using Pascal's triangle:

So, the number of subsets of \(\text{P}\) that contain exactly 3 elements is 20.

##### Equal Sets and Equivalent Sets

Two sets A and B are equal if they contain exactly the same elements

Example:

A = {1, 2, 4} and B = {4, 2, 1}

So set A is equal set B (A = B)

Two sets A and B are said to be equivalent if they have the same cardinality i.e. n(A) = n(B)

Example:

A = {a, i, u, e, o} and B = {2, 3, 5, 7, 11}

Set A is equivalent set B (A ∼ B) because n(A) = n(B) = 5

##### Universal Set

A universal set is the set of all elements or members of all related sets

**Example :**

Describe one possible universal set for each of the following sets!

1. {4, 6, 8, 10}

2. {Neptune, Saturn}

3. {cat, dog}

**Solution:**

1. U = the set of even numbers

2. U = the set of planets in our solar system

3. U = the set of pets

##### Disjoint Set and Joint Set

**Disjoint set** is two or more sets which have no elements in common

Example:

\(\text{U} = \lbrace 1, 2, 3, 4, 6, 7, 8, 9\rbrace\)

\(\text{A} = \lbrace 1, 2, 3, 4\rbrace\)

\(\text{B} = \lbrace 6, 7, 8, 9\rbrace\)

Set A and B are called disjoint set

**Joint set** is two or more sets which have elements in common

Example:

\(\text{U} = \lbrace 1, 2, 3, 5, 7, 9, 11 \rbrace\)

\(\text{A} = \lbrace 1, 3, 5, 7, 9, 11\rbrace\)

\(\text{B} = \lbrace 2, 3, 5, 7\rbrace\)

Set A and B is joint set because they have elements in common {3, 5, 7}

##### Exercise