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If A and B are subsets of a set X, then {A ∩ (X - B)} ∪ B is equal to

If A and B are subsets of a set X, then {A ∩ (X - B)} ∪ B is equal to Correct Answer A ∪ B

Concept:

  • Associative Laws: For any three sets A, B and C, we have
  1. (A ∪ B) ∪ C = A ∪ (B ∪ C)
  2. A ∩ (B ∩ C) = (A ∩ B) ∩ C
  • Distributive Laws: For any three sets A, B and C, we have
  1. A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
  2. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

 

Calculation:

Given: A and B are subsets of a set X

As we know that, X - B = X ∩ B’

⇒ A ∩ (X - B) = A ∩ (X ∩ B’)

⇒ A ∩ (X ∩ B’) = (A ∩ X) ∩ B’ -------------(Associative Law)

⇒ A ∩ (X ∩ B’) = (A ∩ X) ∩ B’ = A ∩ B’ ---------------(A ⊆ X)

⇒ A ∩ (X - B) = A ∩ B’

⇒ {A ∩ (X - B)} ∪ B = (A ∩ B’) ∪ B = (A ∪ B) ∩ (B’ ∪ B) ------------(Distributive Law)

 ⇒ {A ∩ (X - B)} ∪ B = (A ∪ B) ∩ X -------------(∵ B’ ∪ B = X)

⇒ {A ∩ (X - B)} ∪ B = (A ∪ B) ∩ X = A ∪ B ------------(∵ A and B are subsets of a set X)

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