# Equal and Equivalent set

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.