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For any two sets A and B, A - (A - B) equals

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UP TGT Mathematics 2021 Official Paper
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  1. A - B
  2. B
  3. A ∪ B
  4. A ∩ B

Answer (Detailed Solution Below)

Option 4 : A ∩ B
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Detailed Solution

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Concept:

  • P - Q = P ∩ QC      ----(1)
  • De Morgan's Laws: (P ∪ Q)C = PC ⋂ QC and (P ⋂ Q)C = PC ∪ QC      ----(2)
  • Law of Double complementation: (PC)C = P      ----(3)

  • Distributivity: P ⋂ (Q ∪ R) = (P ⋂ Q) ∪ (P ⋂ R)      ----(4)

Calculation:

A - (A - B)

⇒ A - (A ∩ BC)      [using (1)]

⇒ A ∩ (A ∩ BC)C      [using (1)]

⇒ A ∩ (AC ∪ B)      [using (2) & (3)]

⇒ (A ∩ AC) ∪ (A ∩ B)      [using (4)]

⇒ A ∩ B

Hence, A - (A - B) = A ∩ B.

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