# Class-8 Set Operation

Some Facts About Cardinal Number of Set

## Introduction to Set Operation

We have already learnt about the basic concept of sets, representing a set by roster or tabular form or by using set builder form, equal set, finite set, infinite sets, and empty set, cardinal number of a finite set, universal set , equivalent sets and sub sets in previous classes. Now we will study about operation on set.

- Union and intersection of sets
- Difference of two sets
- Complement of a set
- Overlapping and disjoint sets
- Basic result about set's cardinal number

## Union of Sets

The union of two sets **P** and **Q** is the set consisting of all those elements which belong to either **P** or **Q** or both. It is written as **(P ∪ Q)**.

Thus, P ∪ Q = {x|x ∈ P or x ∈ Q}

**Example.** P = {1, 2, 3, 4, 5}

Q = {2, 3, 8, 9}

Then, (P ∪ Q) = {1, 2, 3, 4, 5, 8, 9}

## Intersection of Sets

The set consisting of all those elements which belong to both set **P** and **Q** and it is called intersection of two sets **P** and **Q**. It is written as **P ∩ Q**. It is read as P intersection Q.

Hence, P ∩ Q = { x| x ∈ P and x ∈ Q}

**Example.** If P = {a, e, f, g} and Q = {e, f, h, i)

Then, P ∩ Q = {e, f}

## Difference of Two Sets

Difference of two set consist of all those elements which belong to one set but do not belong to other. It is written as **P − Q**.

Hence, P − Q = {x|x ∈ P and x ∉ Q}

Similarly, for Q − P = {x|x ∈ Q and x ∉ P}

So, P − Q is the set consisting of element of P only and Q − P is the set of consisting of elements of Q only.

**Example.** If P = {0, 4, 5, 6, 7} and Q = {4, 5, 9, 10, 11}

P − Q = {0, 6, 7}, here 0, 6, 7 are elements present in set P only and not present in Q.

Q − p = {9, 10, 11}, here 9, 10, 11 are elements present only in set Q and not present in set P

## Complement of Set

If P is any set and U is the universal set then the complement of P is the set consisting of all the elements of universal set and do not belong to set P. It is written as P'.

Thus P' = {x|x ∈ ∪ and x ∉ P}

**Example.** If P = {1, 2, 3, 6, 9} and U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

P' = {4, 5, 7, 8, 10}

## Overlapping Set

Two set said to be overlapping set, if they have at least one element is common. In other way, two sets are overlapping set if P ∩ Q ≠ ∮

**Example.** Set P = {2, 4, 5, 8, 9} and Set Q = {1, 4, 5, 8, 10, 11}

Here, set P and set Q are overlapping sets because they have elements 4, 5, and 8 are common. So, here P ∩ Q = {4, 5, 8} ≠ ∮

## Disjoint Set

If two sets have no common element, then it is said to be disjoint set. Two set P and Q are disjoint if P ∩ Q = ∮

**Example.** The set P = {February, April, may, June} and Set Q = {Sunday, Monday, Tuesday, Wednesday}

Here, these two sets are disjoint set because they have no common element.

## Properties of Set Operation

- Identity Property
- Idempotent Property
- Complement Property
- Commutative Property
- Associative property
- Distributive property

## Identity Property

If P is any set, then

P ∪ ∅ = P

P ∩ U = P

## Idempotent Property

If P is any set, then

P ∩ P = P

P ∪ P = P

## Complement Property

P ∪ P' = U

P ∩ P' = ∅

## Commutative Property

Intersection and union of sets follow the commutative property.

A ∩ B = B ∩ A

A ∪ B = B ∪ A

## Associative Property

Intersection and union of sets follow the associative property.

(A ∩ B) ∩ C = A ∩ (B ∩ C)

(A ∪ B) ∪ C = A ∪ (B ∪ C)

## Distributive Property

Intersection and union of sets follow the distributive property.

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Let's see some solved examples to understand the properties better.

**Example 1.**If P = {4, 6, 8, 11, 13, 16, 20 ,22, 25}

Q= { 7, 9 , 11,16, 18 , 22,25}

R = { 0, 5, 9, 18,20, 25}

Find i) P ∪ Q

ii) P ∪ R

iii) P ∩ Q

iv) Q ∩ R

v) Q ∪ R

VI) P ∩ R

**Solution.** i) P ∪ Q ={4, 6, 8, 11, 13, 16, 20, 22, 25, 7, 9, 18, 20}

ii) P ∪ R = {4, 6, 8, 11, 13, 16, 20, 22, 25, 0, 5, 9, 18)

iii) P ∩ Q = {11, 16, 22, 25}

iv) Q ∩ R = {9, 18, 25}

v) Q ∪ R = {7, 9, 11, 16, 18, 22, 25, 0, 5, 20}

vi) P ∩ R = {20, 25}

**Example 2.** If universal set U = {10, 20, 30, 40, 50, 60, 70, 80, 90} and set A = { 40, 60, 80, 90}, then find A'.

**Solution.** Complement of A = A' ={10, 20, 30, 50, 70}

**Example 3.** If Set A = {BHUBANESWAR} and Set B = {AHMEDABAD},

Find i) A ∪ B ii) A ∩ B iii) A − B IV) B − A

**Solution** Roster form of set A = {B, H, U, A, N, E, S, W, R}

Roster form of set B = {A, H, M, E, D, B}

i) A ∪ B = {B, H, U, A, N, E, S, W, R, M, D}

ii) A ∩ B = {B, H, A, E}

iii) A − B = {U, N, S, W, R}

iv) B − A = {M, D}

## Some Facts About Cardinal Number of Set

If P and Q are two finite sets, then

- n(P ∪ Q) = n(P) + n(Q) − n(P ∩ Q)
- n(P − Q ) = n(P ∪ Q) − n(Q) = n(P) − n(P ∩ Q)
- n(Q − P ) = n(P ∪ Q) − n(P)= n(Q) − n(P ∩ Q)
- n(P ∪ Q) = n(P − Q) + n(Q − P) + n( P ∩ Q)

**Example 1.** If n(P) = 18, n(Q) = 14, and n(P ∪ Q) = 20 ,Then find n(P ∩ Q)

**Solution.** n(P ∪ Q) = n(P) + n(P ) − n(P ∩ Q)

=> 20 = 18 + 14 − n(P ∩ Q)

=> n(P ∩ Q) = 32 − 20 = 12

**Example 2.** n(U) = 30, n(P') =12, n(Q) =15, n(P ∩ Q) = 10, Find

i) n(P')

ii) n(P U Q)

**Solution.**

i) n(Q') = n(U) − n(Q)

= 30 − 15 = 15

ii) n(P ∪ Q) = n(P) + n(Q) − n( P ∩ Q)

here, we first find out n(P) by using formulae

n(P') = n(U) − n(P)

=> 12 = 30 − n(P)

=> n(P) = 18

Then, we put the value of n(P) below equation

n(P ∪ Q) = 18 + 15 − 10=23

**Example 3.** If n(U)= 50, n(P)= 30, n(Q)= 10 and n{(P ∪ Q)'} = 18, then find

i) n(P ∪ Q )

ii) n( P ∩ Q)

iii) n(P − Q)

**Solution.**

i) We know n(P ∪ Q) + n{(P ∪ Q)'} = n(U)

=> n(P ∪ Q) + 18 = 50

=> n(P ∪ Q) = 50 − 18 = 32

ii) n(P ∪ Q) = n(P) + n(Q) − n(P ∩ Q)

=> 32 = 30 + 10 − n(P ∩ Q)

=> n(P ∩ Q) = 40 − 32 = 8

iii) n(P − Q ) = n(P) − n(P ∩ Q)

=> n(P − Q ) = 30 &minus 8 = 22

## Class-8 Set Operation Worksheets

## Answer Sheet

**Set-Operation-Answer**Download the pdf

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