**Operations
of Sets**

**We can perform operations on two given sets such as union of sets, intersection of sets and difference of sets. The operations of sets concept is very important in the field of set theory. Let us see these operations of sets one by one.**

** Union
of Sets**

** Intersection
of Sets**

** Difference
of Sets**

*x*:

*x*∈ A,

*x*∉ B} and B – A = {

*x*:

*x*∈ B,

*x*∉ A}.

** Venn
Diagram**

** Venn Diagram of Union
of Sets**

** ****Venn Diagram of ****Intersection
of Sets**

## **Venn Diagram of ****Difference
of Sets**

**Properties of Operations on Sets**

**Distributive Property **

1. If A, B and C are three sets,
then A ∪ (B ∩ C) = (A ∪ B) ∩
(A ∪ C)

This property is called
distributive property over the intersection of two sets.

2.
If A, B and C are three sets, then the intersection is distributive over union
of two sets, i.e., A ∩ (B ∪
C) = (A ∩
B) ∪ (A ∩ C)

**De Morgan’s Laws **

If A and B
are two sets, then De Morgan’s laws state that:

1. (A ∪ B)′ = A′ ∩ B′

2. (A ∩
B)′
= A′ ∪ B′

To verify the above laws, let us consider an example.

If U = {1, 2, 3,
… , 20}, A = {5, 10, 15, 20} and B = {4, 8, 12, 16, 20}, then A ∩ B = {20}

(A ∩ B)′ = {1,
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}

A′ = {1, 2, 3,
4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19}

B′ = {1, 2, 3,
5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19}

A′ ∪ B′ = {1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}

∴ (A ∩ B)′ = A′ ∪ B′

Now, A ∪ B = {4, 5, 8, 10, 12, 15, 16, 20}

(A ∪ B)′ = {1, 2, 3, 6, 7, 9,
11, 13, 14, 17, 18, 19}

A′ ∩ B′ = {1,
2, 3, 6, 7, 9, 11, 13, 14, 17, 18, 19}

∴ (A ∪ B)′ = A′ ∩ B′

**Cardinal Properties of Sets **

1. If A and B
are two sets, then *n*(A ∪ B) = *n*(A) + *n*(B)
– *n*(A ∩ B).

If A and B are
disjoint sets, then A ∩ B = φ. Hence, *n*(A ∩ B) = 0.

∴ For disjoint sets, *n*(A ∪ B) = *n*(A) + *n*(B).

2. *n*(A
– B) = *n*(A ∪ B) – *n*(B) = *n*(A)
– *n*(A ∩ B)

3. *n*(B
– A) = *n*(A ∪ B) – *n*(A) = *n*(B)
– *n*(A ∩ B)

4. *n*(A ∪ B) = *n*(A – B) + *n*(B – A) + *n*(A ∩ B)

5. If
universal set (U) is finite and A is any set, then *n*(A) + *n*(A′) =
*n*(U)

**Example 1: **If *n*(A) = 40, *n*(B) = 27 and *n*(A ∩
B) = 15, find

a. *n*(A ∪ B)

b. *n*(B
– A)

c. *n*(only
B)

** **

**Solution: **

a. We know
that, *n*(A ∪ B) = *n*(A) + *n*(B) – *n*(A ∩ B)

∴ *n*(A ∪ B) = 40 + 27 – 15 ⇒ *n*(A ∪ B) = 52

b. *n*(B
– A) = *n*(A ∪ B) – *n*(A) = 52 – 40
= 12

c. *n*(only
B) = *n*(B – A) = 12

** **

**Example 2: **If *n*(A – B) = 12, *n*(B – A) = 16 and *n*(A
∩ B) = 5, find

a. *n*(A)

b. *n*(B)

c. *n*(A ∪ B)

** **

**Solution: **

a. We know
that, *n*(A – B) = *n*(A) – *n*(A ∩ B)

∴ 12 = *n*(A) – 5 ⇒ *n*(A) =
17

b. We know
that, *n*(B – A) = *n*(B) – *n*(A ∩ B)

∴ 16 = *n*(B) – 5 ⇒ *n*(B) =
21

c. Again, *n*(B
– A) = *n*(A ∪ B) – *n*(A)

∴ 16 = *n*(A
∪ B) – 17 ⇒ *n*(A ∪ B) = 33