Question

If A = {x : x is a multiple of 3}, B = {x : x is a multiple of 4} and C = {x : x is a multiple of 12}, then A ∩ (B ∩ C) is equal to
1. A
2. B
3. C
4. {x : x is a multiple of 144}

Answer 1

Correct Answer - Option 3 : C

Concept:

     If A, B, and C are subsets of a set X. Then

• Commutative property: A ⋃ B = B ⋃ A
• Associative property: (A ⋃ B) ∪ C = A ⋃ (B ⋃ C)
• Distributive property: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) or A ⋂ (B ⋃ C) = (A ⋂ B) ⋃ (A ⋂ C)

 

Calculation:

The given problem is,

A = {x : x is a multiple of 3}

B = {x : x is a multiple of 4}

C = {x : x is a multiple of 12}

If we look closely at the set C, we will observe that C = A ∩ B

This is because L.C.M of (3,4) is 12

Now, let's come back to the question

A ∩ (B ∩ C) = A ∩ (B ∩ (A ∩ B)) , since C = A ∩ B

= A ∩ (A ∩ B)

= A ∩ B

= C

Hence, A ∩ (B ∩ C) = C