# Operation on Sets | Union of sets | Intersection of sets | Difference of sets

In this article **Operation on Sets** we give the information about **Union of sets**, Intersection of sets, **Difference of sets**. also give **DeMorgan’s laws**, Union of sets examples, Difference of sets examples.

## Operation on Sets:

**Union of two sets: **

Let P and Q be two given sets. Then the set of all elements of set P and Q is called the Union of two sets. It is written as P ∪ Q and read as ‘P union Q’.

**Note:** common elements you can take only one time.

P ∪ Q = {x | x ⊂ P or x ⊂ Q}

**Ex. 1)** P = {0, 1, 2, 3, 4, 5 }

Q = {0, 5, 6, 7, 8, 9}

P ∪ Q = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

Note that, P ∪ Q = Q ∪ P

This can be represented by Venn diagram as shown below: –

**Ex (2)** Observe the Venn diagram and write the following sets using listing method.

1) U 2) A 3) B 4) A ∪ B 5) A ∩ B 6) A’ 7) B’ 8) (A ∪ B)’ 9) (A ∩ B)’

**Solution :**

U = { -6, -2, 0, 1, 3, 4, 5, 7, 8, 9, 10, 12, 15 }

A = { -2, 0, 1, 4, 7, 9, 10 }

B = { -6, -2, 0, 1, 3, 5 }

A ∪ B ={ -6, -2, 0, 1, 3, 4, 5, 7, 9, 10}

A ∩ B = { -2, 0, 1}

A’ = { -6, 3, 5, 8, 12, 15}

B’ = { 4, 7, 8, 9, 10, 12, 15 }

(A ∪ B)’ ={ 8, 12, 15 }

(A ∩ B)’ = { -6, 3, 4, 5, 7, 8, 9, 10, 12, 15 }

#### Intersection of two sets:

Suppose P and Q are two sets. The set of all common elements of P and Q is called the intersection of set P and Q. It is denoted as P ∩ Q and read as P intersection P.

Therefor, P ∩ Q = {x | x ⊂ P and x ⊂ Q}

**Ex (1)** P = { 1, 2, 3, 4, 5, 7 } Q = { 1, 4, 6, 7, 8, 10 } Let us draw Venn diagram.

The elements 1, 4 and 7 is common in set P and Q.

Therefor, P ∩ Q = { 1, 4, 7 }

**Ex. (2) **P = { 1,2, 3, 6, 7, 8, 9, 11, 15, 17 } Q = { 1, 6, 11}

The elements 1, 6, 11 are common in set P and Q

Therefor, P ∩ Q = { 1, 6, 11 } but Q = { 1, 6, 11 }

Therefor, P ∩ Q = Q

Here set Q is the subset of P.

therefor, If Q ⊆ P then P ∩ Q = Q, similarly, if Q ∩ P = Q, then Q ⊆ P

**Set Difference: **

The relative complement or set difference of sets P and Q, denoted P – Q, is the set of all elements in P that are not in Q.

In set-builder notation, P – Q = {x ∈ U : x ∈ P and x ∉ Q}= P ∩ Q’.

The Venn diagram for the set difference of sets P and Q is shown below:

**Similarly:**

The relative complement or set difference of sets Q and P, denoted Q – P, is the set of all elements in Q that are not in P.

In set-builder notation, Q – P = {x ∈ U : x ∈ Q and x ∉ P}= Q ∩ P’.

The Venn diagram for the set difference of sets Q and P is shown below:

**Difference of sets Q and P**

**Ex. 1) ** X = {1, 2, 3, 4, 5, 6,7, 8, 9} and Y = {14, 15, 16, 11, 12, 13}.

Find the difference between the two sets:

(i) X and Y

(ii) Y and X

**Solution:**

The two sets are disjoint as they do not have any elements in common.

(i) X – Y = { 1, 2, 3, 4, 5, 6, 7, 8, 9} = X

(ii) Y – X = { 14, 15, 16, 11, 12, 13} = Y

**Ex. 2)** Let M = {a, b, c, d, e, f, g, h, i, j, k} and N = {b, d, f, g, x, y, z}.

Find the difference between the two sets:

(i) M and N

(ii) N and M

**Solution:**

(i) M – N = { a, c, e, h, i, j, k }

Therefore, the elements a, c, e, h, i, j, k belong to M but not to N

(ii) N – M = { x, y, z}

Therefore, the elements x, y, z belongs to N but not M.

**Ex. 3)** Given three sets P, Q and R such that:

P = {x : x is a natural number between 10 and 18},

Q = {y : y is a even number between 8 and 24} and

R = {7, 9, 11, 14, 18, 20, 22, 24}

(i) Find the difference of two sets P and Q

(ii) Find Q – R

(iii) Find R – P

(iv) Find Q – P

**Solution:**

According to the given statements:

P = {11, 12, 13, 14, 15, 16, 17}

Q = {10, 12, 14, 16, 18, 20, 22}

R = {7, 9, 11, 14, 18, 20, 22, 24}

(i) P – Q = {Those elements of set P which are not in set Q}

= {11, 13, 15, 17}

(ii) Q – R = {Those elements of set Q not belonging to set R}

= {10, 12, 16}

(iii) R – P = {Those elements of set R which are not in set P}

= {7, 9, 18, 20, 22, 24}

(iv) Q – P = {Those elements of set Q not belonging to set P}

= {10, 18, 20, 22}

#### De Morgan’s Laws (without proof).

**( P ∩ Q )’ = P’ ∪ Q’****( P ∪ Q )’ = P’**∩ Q’

**Related Links :**

**complement of a set | types of set theory | Roster method | universal set****Age problems tricks || word problems ||age problems questions****average meaning || average problems || average definition****Boat and stream questions || boats and streams tricks**