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.

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