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: –

Union of set
Union of set

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

Union example

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 }

Intersection Example

 

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

Intersection Ex.

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:

difference of sets P and Q
Difference of sets P and Q

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

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).

  1. ( P ∩ Q )’ = P’ ∪ Q’
  2. ( P ∪ Q )’ = P’ ∩ Q’

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Santosh Nalawade

Work as Assistant Professor and Web Developer.

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