# Union, Intersection, Compliment, Difference of Sets ## Operations On Sets

In earlier classes, we learned how to perform the operations of addition, subtraction, multiplication, and division on numbers. Each one of these operations was performed on a pair of numbers to get another number. For example, when we perform the operation of addition on the pair of numbers 5 and 13, we get the number 18. Again, performing the operation of multiplication on the pair of numbers 5 and 13, we get 65. Similarly, there are some operations that when performed on two sets give rise to another set. We will now define certain operations on sets and examine their properties. Henceforth, we will refer to all our sets as subsets of some universal set.

## Union of Sets

Lets A and B be any two sets. The union of A and B is the set that consists of all the elements of A and all the elements of B, the common elements being taken only once. The symbol ‘∪’ is used to denote the union. Symbolically, we write A ∪ B and usually read as ‘A union B’.

Definition – The union of two sets A and B is set C which consists of all those elements which are either in A or in B (including those which are in both). In symbols, we write.

A ∪ B = { x : x ∈ A or x ∈ B } Some Properties of the Operation of Union

(i) A ∪ B = B ∪ A (Commutative law)
(ii) ( A ∪ B ) ∪ C = A ∪ ( B ∪ C) (Associative law )
(iii) A ∪ φ = A (Law of identity element, φ is the identity of ∪)
(iv) A ∪ A = A (Idempotent law)
(v) U ∪ A = U (Law of U)

Example – Let A = { 2, 4, 6, 8} and B = { 6, 8, 10, 12}. Find A ∪ B.

Solution:- We have A ∪ B = { 2, 4, 6, 8, 10, 12} Note that the common elements 6 and 8 have been taken only once while writing A ∪ B.

## Intersection of Sets

The intersection of sets A and B is the set of all elements which are common to both A and B. The symbol ‘∩’ is used to denote the intersection. The intersection of two sets A and B is the set of all those elements which belong to both A and B. Symbolically, we write A ∩ B = {x : x ∈ A and x ∈ B}.

Example –  Consider sets A and B of Example 12. Find A ∩ B.

Solution:- We see that 6, and 8 are the only elements that are common to both A and B.
Hence A ∩ B = { 6, 8 }.

Definition – The intersection of two sets A and B is the set of all those elements which belong to both A and B. Symbolically, we write

A ∩ B = {x : x ∈ A and x ∈ B} Some Properties of Operation of Intersection

(i) A ∩ B = B ∩ A (Commutative law).
(ii) ( A ∩ B ) ∩ C = A ∩ ( B ∩ C ) (Associative law).
(iii) φ ∩ A = φ, U ∩ A = A (Law of φ and U).
(iv) A ∩ A = A (Idempotent law)
(v) A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) (Distributive law ) i. e., ∩ distributes over ∪. ## Difference of Sets

The difference between the sets A and B in this order is the set of elements that belong to A but not to B. Symbolically, we write A – B and read as “ A minus B” this process is the difference of Sets.

Difference of Sets

Example – Let A = { 1, 2, 3, 4, 5, 6}, B = { 2, 4, 6, 8 }. Find A – B and B – A.

Solution – We have, A – B = { 1, 3, 5 }, since the elements 1, 3, 5 belong to A but not to B, and B – A = { 8 } since element 8 belongs to B and not to A.

###### Difference of Sets

We note that A – B ≠ B – A. ## Complement of A Sets

Let U be the universal set which consists of all prime numbers and A be the subset of
U which consists of all those prime numbers that are not divisors of 42. Thus, A = {x : x ∈ U and x is not a divisor of 42 }. We see that 2 ∈ U but 2 ∉ A because 2 is the divisor of 42. Similarly, 3 ∈ U but 3 ∉ A, and 7 ∈ U but 7 ∉ A. Now 2, 3, and 7 are the only elements of U which do not belong to A. The set of these three prime numbers, i.e., the set {2, 3, 7} is called the Complement of A with respect to U.

Some Properties of Complement Sets

1. Complement laws: (i) A ∪ A′ = U (ii) A ∩ A′ = φ
2. De Morgan’s law: (i) (A ∪ B)´ = A′ ∩ B′ (ii) (A ∩ B)′ = A′ ∪ B′
3. Law of double complementation : (A′)′ = A
4. Laws of empty set and universal set φ′ = U and U′ = φ.

These laws can be verified by using Venn diagrams.

### Complement of the union of sets

The complement of the union of two sets is the intersection of their complements and the complement of the intersection of two sets is the union of their complements. These are called De Morgan’s laws. # Practical Problems on Union and Intersection of Two Sets

In an earlier Section, we learned about union, intersection, and difference between two sets. In this Section, we will go through some practical problems related to our daily life.  