In this chapter we will learn about the definition of equal and equivalent sets with properties and solved examples.

## Equal sets

**Two sets A & B are equal when both the sets have same elements**.

This means that every element of set A is a member of B and every element of set B is member of A.

**Note:**

The arrangement of elements in the set is not important.

We have to look if all the elements in both set are similar or not.

### Representing equal sets

If two sets A & B are equal then the relationship can be expressed as;**A = B**

The expression signifies that both the elements have same elements.

### Examples of Equal set

(i) A = { 2, 5, 7, 9 }

B = { 7, 2, 5, 9 }

**Explanation**

Both the sets A & B are equal since all elements of set A are also part of set B and vice- versa.

Note that the elements in the given set can be listed in any order.

(ii) **A = { x : x is letter in word ” LOYAL ” }****B = { x : x is letter in word ” ALLOY ” }**

**Explanation**

Representing both the sets in Roster form.

A = { x : x is letter in word ” LOYAL ” }

Roster form of set A is;

A = { L, O, Y, A }

Here we have removed the repeated elements.

Roster form of set B is;

B = { A, L, O, Y }

Both the sets are equal since they contain same elements.

Hence, A = B

(iii) **P = {1, 2, 3}****Q = {1, 2, 2, 3}**

Check if the above two sets are equal.**Explanation**

The two given sets are;

P = {1, 2, 3}

Q = {1, 2, 2, 3}

Removing the repeated elements from set Q.

Q = {1, 2, 3}

Both the sets P & Q have the same elements, hence they are equal.

P = Q

(iv)** A = { x : x is a letter of word “WOLF” }****B = { x : x is a letter of word “FOLLOW” }**

**Solution**

Writing the above sets in Roster form.

A = { W, O, L, F }

B = { F, O, L, W }

Observe that all elements of set A are also part of set B and vice-versa.

Hence, A = B.

## Equivalent Set

**Two sets are equivalent when their order number is same.**

Hence, the equivalent sets have **same number of unique elements**.

### Representing Equivalent set

If two sets A & B are equivalent then the relationship can be expressed as;

**n (A) = n (B)**

Where n(A) & n(B) tells the cardinal number of sets.

### Examples of Equivalent set

(i) Given below are two sets A & B. Check if the sets are equivalent or not.**A = { 1, 2, 3 }****B = { a, b, c }****Solution**

Let’s find the cardinal number of both the sets.

A = {1, 2, 3}

There are three unique elements in set A.

So the cardinal number of set A is 3.

n(A) = 3

B = { a, b, c }

There are three unique elements in set B.

So the cardinal number of set B is 3.

n(B) = 3

You can see that n(A) = n(B) = 3.

Hence both the sets A & B are equivalent set.

(ii) Given below are four sets A, B, C & D. Pair the equivalent sets.**A = { 5, 7, 10, 13 }****B = { l, m, r }****C = { x, y, z, b, z, b, x }**

**Solution**

Let’s find the cardinal number of each of the sets.**A = { 5, 7, 10, 13 }**

There are 4 unique elements in the set.**n (A) = 4**

**B = { l, m, r } **

There are 3 unique element in set B.**n (B) = 3**

**C = { x, y, z, b, z, b, x }**

Removing the repeating elements.

C = {x, y, z, b}

There are three unique elements in set C.**n (C) = 4**

In the above three sets, set A & C have the same order.

n(A) = n(C) = 4

Hence both the sets are equivalent sets.