## Set Operations

Till now the students have dealt with mainly four basic operations of mathematics i.e. addition, subtraction, multiplication and division. These operations are mainly applied to two or more numbers to obtain a result that is a combination of these numbers. For example, when we apply the operation of addition on two numbers suppose 7 and 2, we get the number 9. Likewise, set operations are a group of operations that are applied on two or more sets that combines them and results in a single set. The set operations consist of three types of operations namely the union of sets (U), the intersection of sets (⋂) and the difference between sets (-). Let us understand all the set operations with suitable examples:

### Union of Sets

Suppose A and B are sets consisting of elements. Let's say set A = {4, 5, 6, 2, 1} and set B = {7, 8, 9, 0}.

A⋃B is read as ‘A union B’, that means ‘union’ is denoted as ‘⋃’.

Therefore, A ⋃ B = {4, 5, 6, 2, 1} ⋃ {7, 8, 9, 0}.

= {4, 5, 6, 2, 1, 7, 8, 9, 0}

Hence, we see that the union of sets A and B consists of all the elements that were in set A and set B respectively.

### Intersections of Sets

Suppose A and B are sets consisting of elements. Let's say set A = {8, 9, 5, 4, 6, 2} and set B = {5, 2, 3, 1, 9}.

A∩B is read as ‘A intersection B’, that means ‘intersection’ is denoted as ‘∩’.

Therefore, A⋂B = {8, 9, 5, 4, 6, 2}⋂{5, 2, 3, 1, 9}.

= {5, 2, 9}

Hence, when we apply intersection on two sets it gives us the elements that are common or are a part of both the sets.

### Difference of Sets

Suppose A and B are two sets consisting of elements. Let’s say set A = {5, 6, 8, 9, 0} and B = { 9, 6, 0, 7, 3}

A - B is read as ‘A minus B’, that means the difference or minus is denoted as ‘-’.

Therefore, A - B = {5, 6, 8, 9, 0} - { 9, 6, 0, 7, 3}

= {5,8}

We can see that the difference of two sets A and B results in the set of elements that are a part of set A but are not in set B.

Similarly, B - A = { 9, 6, 0, 7, 3} - {5, 6, 8, 9, 0}

= {7, 3}

Here also, B - A results in the set of elements that are a part of set B but are not in set A.

### Solved Examples

1. If A = {6, 9, 8, 1}, B = {1, 7, 5, 2, 9}, C = {7, 6, 0, 1} and D = {0, 1}. Find:-

A ∩ B

B ∩ C

A ∩ C

B ∩ D

A ∩ D

A ∩ (B U C)

A ∩ (B U D)

Answer - (a) A ∩ B = {6, 9, 8, 1} ∩ {1, 7, 5, 2, 9}

= {1, 9}

(b) B ∩ C = {1, 7, 5, 2, 9} ∩ {7, 6, 0, 1}

= {1, 7}

(c) A ∩ C = {6, 9, 8, 1}⋂ {7, 6, 0, 1}

= {1, 6}

(d) B ∩ D = {1, 7, 5, 2, 9} ⋂ {0, 1}

= {1}

(e) A ∩ D = {6, 9, 8, 1} ⋂ {0, 1}

= {1}

(f) A ∩ (B U C) = {6, 9, 8, 1} ⋂ ({1, 7, 5, 2, 9} U {7, 6, 0, 1})

= {6, 9, 8, 1} ∩ {1, 7, 5, 2, 9} U {6, 9, 8, 1} ⋂ {7, 6, 0, 1}

= {1, 9} U {1, 6}

= {1, 9, 6}

(g) A ∩ (B U D) = {6, 9, 8, 1} ∩ ({1, 7, 5, 2, 9} U {0, 1})

= {6, 9, 8, 1} ∩ {1, 7, 5, 2, 9} U {6, 9, 8, 1} ∩ {0, 1}

= {1,9} U {1}

= {1, 9}

## FAQs on What are Set Operations?

1. What are the Properties of Set Operations?

Answer: The three main properties of operation of sets are:

**Commutative Property**

AUB = BUA (commutative property of Union).

A∩B = B∩A (commutative property of intersection).

**Associative Property**

(AUB)UC = AU(BUC) (Associative property of union)

(A∩B)∩C = A∩(B∩C) (Associative property of intersection)

**Distributive Property**

AU(B∩C) = (AUB) ∩ (AUC)

A⋂(BUC) = (A∩B) U (A⋂C)

Some of the other properties of set operations are as follows:-

**Law of U and Φ(Phi)**

A U Φ = A

Φ ⋂ A = Φ

**Idempotent Law**

A U A = A

A ∩ A = A

**Law of U(Universal Set)**

U U A = U

U ⋂ A = A

2. What are Disjoint Sets? Give Two Examples.

Answer: When two sets do not have any common elements such that A ∩ B = Φ, they are called disjoint sets.

The examples of disjoint sets are as follows:

Set A = {8, 9, 3, 4} and Set B = {1, 7, 6}

As we can see, set A and set B do not have common elements.

Therefore, A ⋂ B = {8, 9, 3, 4} ⋂ {1, 7, 6}

= Φ

Set P = {a, b, c, d, e, f} and Set = {g, h, i, j, k, l}

Therefore, P ∩ Q = {a, b, c, d, e, f} ∩ {g, h, i, j, k, l}

= Φ