Properties of Intersection of Sets

Different properties of the Intersection of Sets are:

• Commutative Law
• Associative Law
• Distributive Law
• Law of Empty Set (ϕ)
• Law of Universal Set (U)
• Idempotent Law

Let’s discuss the above properties in detail. Consider three sets A, B, and C.

Commutative Law for Intersection

The commutative law of the intersection of sets states that the ordering of sets does not matter while performing an intersection.

A ∩ B = B ∩ A

Example: If A = {1, 10, 24, 31} and B = {1, 7,10, 24}. Verify commutative law.

Solution:

Given:

A = {1, 10, 24, 31}

B = {1, 7,10, 24}

A ∩ B = {1, 10, 24, 31} ∩ {1, 7,16, 24}

⇒ A ∩ B = {1, 24} —–(I)

B ∩ A = {1, 7,16, 24} ∩ {1, 10, 24, 31}

⇒ B ∩ A = {1, 24} —–(II)

From (I) and (II)

A ∩ B = B ∩ A [Hence Verified]

Associative Law for Intersection

The associative law of the intersection of sets states that we can perform the intersection of any two sets first in any order and then after other sets.

(A ∩ B) ∩ C = A ∩ (B ∩ C)

Example: If P = {2, 7, 15}, Q = {2, 9, 13}, R = {2, 23} then verify associative law.

Solution:

Given:

P = {2, 7, 15}, Q = {2, 9, 13}, R = {2, 23}

(P ∩ Q) ∩ R = [{2, 7, 15} ∩ {2, 9, 13}] ∩ {2, 23}

⇒ (P ∩ Q) ∩ R = {2} ∩ {2, 23}

⇒ (P ∩ Q) ∩ R = {2} ———(I)

Now,

P ∩ (Q ∩ R) = {2, 7, 15} ∩ [{2, 9, 13} ∩ {2, 23}]

⇒ P ∩ (Q ∩ R) = {2, 7, 15} ∩ {2}

⇒ P ∩ (Q ∩ R) = {2} ———(II)

From (I) and (II)

(P ∩ Q) ∩ R = P ∩ (Q ∩ R) [Hence Verified]

Distributive Law for Intersection

The distributive law of the intersection of sets states that the intersection of a set A with the union of the other two sets B and C is equivalent to the intersection of set A and B union intersection of set A and C.

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Example: If P = {a, b, d}, Q = {b, c, d}, R = {b, d, j} then verify associative law.

Solution:

Given:

P = {a, b, d}, Q = {b, c, d}, R = {b, d, j}

⇒ (P ∩ Q) = {a, b, d} ∩ {b, c, d} = {b, d}

⇒ (P ∩ R) = {a, b, d} ∩ {b, d, j} = {b, d}

(Q ∪ R) = {b, c, d} ∪ {b, d, j} = {b, c, d, j}

⇒ P ∩ (Q ∪ R) = {a, b, d} ∩ {b, c, d, j}

⇒ P ∩ (Q ∪ R) = {b, d} ———(I)

Now,

(P ∩ Q) ∪ (P ∩ R) = {b, d} ∪ {b, d}

⇒ (P ∩ Q) ∪ (P ∩ R) = {b, d} ———(II)

From (I) and (II)

P ∩ (Q ∪ R) = (P ∩ Q) ∪ (P ∩ R) [Hence Verified]

Law of Empty set (ϕ) for Intersection

Law of empty set ϕ states that the intersection of an empty set with any other set results in the empty set.

A ∩ ϕ = ϕ

Example: If A = {1, 32} then verify the law of the Empty set.

Solution:

Given,

A = {1, 32}

⇒ A ∩ ϕ = {1, 32} ∩ { }

⇒ A ∩ ϕ = { }

Thus, A ∩ ϕ = ϕ [Hence Verified] 

Law of Universal set (U) for Intersection

Law of Universal set U states that the intersection of a universal set with any set results in the same set.

A ∩ U = A

Example: If A = {10, 12} and U = {10, 12, 14, 16} then verify the law of the Universal set.

Solution:

Given,

A = {10, 12} and U = {10, 12, 14, 16}

⇒ A ∩ U = {10, 12} ∩ {10, 12, 14, 16}

⇒ A ∩ U = {10, 12}

Thus, A ∩ U = A [Hence Verified]

Idempotent Law

Idempotent law states that the intersection of two same sets results in the same set.

A ∩ A = A

Example: If A = {5, 20} then verify Idempotent law.

Solution:

Given, A = {5, 20}

A ∩ A = {5, 20} ∩ {5, 20}

⇒ A ∩ A = {5, 20}

⇒ A ∩ A = A [Hence Verified]

Intersection of Sets is the operation in set theory and is applied between two or more sets. It result in the output as all the elements which are common in all the sets under consideration. For example, The  intersection of sets A and B is the set of all elements which are common to both A and B.

In other words, it is an operation that selects the common identical element among the sets. For example

Suppose, Set A is the set of odd Numbers less than 10 and Set B is the set of first 5 multiple of 3.

⇒ A = {1,3,5, 7,9}
⇒ B = {3,6,9,12,15}

So the common element in these two set are 3 and 9.

Hence, the set of intersection of A and B = {3,9}