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A, B, C and D are four sets such that A ∩ B = C ∩ D = ϕ. Consider the following: 1. A ∪ C and B ∪ D are always disjoint. 2. A ∩ C and B ∩ D are always disjoint. Which of the above statements is/are correct?

A, B, C and D are four sets such that A ∩ B = C ∩ D = ϕ. Consider the following: 1. A ∪ C and B ∪ D are always disjoint. 2. A ∩ C and B ∩ D are always disjoint. Which of the above statements is/are correct? Correct Answer 2 only

Concept:

Two sets are said to be disjoint when they have no common element.

Two sets A and B are disjoint sets if the intersection of two sets is a null set or an empty set.

A ∩ B = ϕ

Calculation:

Given: A, B, C and D are four sets such that A ∩ B = C ∩ D = ϕ

Statement 1: A ∪ C and B ∪ D are always disjoint.

⇒ (A ∪ C) ∩ (B ∪ D) = ∪

= ∪

= ∪

= (C ∩ B)] ∪ ∩

= ∩

= ∩                   (∵ A ∩ ϕ = ϕ)

= (ϕ ∩ ϕ)   

= ϕ              

(A ∩ C) and (B ∩ D) are always disjoint.

Hence statement 2 is correct.

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