Set Operations: Intersection and Difference of Two Sets

Intersection of Sets

The intersection of two sets A and B which are subsets of the universal set U, is the set which consists of all those elements which are common to both A and B.
It is denoted by ‘∩’ symbol. All those elements which belong to both A and B represent the intersection of A and B. Thus we can say that,

A ∩ B = {x : x ∈ A and x ∈ B}

For n sets $A_{1},A_{2}, A_{3} , …… A_{n}$ where all these sets are the subset of universal set U the intersection is the set of all the elements which are common to all these n sets.

Depicting this pictorially, the shaded portion in the Venn diagram given below represents the intersection of the two sets A and B.

Figure 1-Intersection of two sets

Figure 2-Intersection of three sets

Intersection of Two sets:

If A and B are two sets, then the intersection of sets is given by:

$A \cap B = n(A) + n (B) – n (A \cup B)$

where n(A) is the cardinal number of set A,
n(B) is the cardinal number of set B,
$n (A \cup B)$ is the cardinal number of union of set A and B.

Example: Let U be the universal set consisting of all the n – sided regular polygons where 5 ≤ n ≤ 9. If set A,B and C are defined as:

A = {pentagon,hexagon,octagon}

B = {hexagon,nonagon,heptagon}

C = {nonagon}

Find the intersection of the sets:

i) A and B

ii) A and C

Solution: U = {pentagon , hexagon , heptagon , octagon , nonagon}

i) The intersection is given by all the elements which are common to A and B.

A ∩ B = {hexagon}

ii)  No element is common in A and C. Therefore A ∩ C = ∅

Note: If we have two sets X and Y such that their intersection gives an empty set ∅ i.e. X ∩ Y = ∅ then these sets X and Y are called as disjoint sets.

Properties of Intersection of a Set:

• i)    Commutative Law: The union of two sets A and B follow the commutative law i.e.,

A ∩ B = B ∩ A

• ii)   Associative Law: The intersection operation follows the associative law i.e., If we have three sets A ,B and C then,

(A ∩ B) ∩ C = A ∩ (B ∩ C)

• iii)  Identity Law: The intersection of an empty set with any set A gives the empty set itself i.e.,

A ∩ ∅ = ∅

• iv)   Idempotent Law: The intersection of any set A with itself gives the set A i.e.,

A ∩ A = A

• v)    Law of U: The intersection of a universal set U with its subset A gives the set A itself.

A ∩ U = A

• vi)   Distributive Law: According to this law:

A ∩ (B ∪ C) = ( A ∩ B ) ∪ (A ∩ C)

Difference of Sets:

Difference of two sets A and B is the set of elements which are present in A but not in B. It is denoted as A-B. In the following diagram the region shaded in orange represents the difference of sets A and B. And the region shaded in violet represents the difference of B and A.

For Example,

Let A = {3 , 4 , 8 , 9 , 11 , 12 } and B = {1 , 2 , 3 , 4 , 5 }. Find A – B and B – A.

Solution: We can say that A – B = {8 , 9 , 11 , 12} as these elements belong to A but not to B

B – A ={1,2,5} as these elements belong to B but not to A.