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Let P and Q be two distinct non-empty sets. Then  (P ∪ Q)c  ∪ (Pc ∩ Q) equals

Let P and Q be two distinct non-empty sets. Then  (P ∪ Q)c  ∪ (Pc ∩ Q) equals Correct Answer P<sup><span style="font-size: 10.5px;">c</span></sup>

Concept:

Law of Union: (P ∪ Q)c = Pc ∩ Qc 

Law of Intersection: (P ∩ Q)c = Pc ∪ Qc

Distributive Property:

  • P ∪ (Q ∩ R) = (P ∩ Q) ∪ (P ∩ R)
  • P ∩ (Q ∪ R) = (P ∪ Q) ∩ (P ∪ R)

 

Calculation:

Set A (say) = (P ∪ Q)c  ∪ (Pc ∩ Q)

By Law of Union 

⇒ A = (Pc ∩ Qc) ∪ (Pc ∩ Q)

Using Distributive property

⇒ A = Pc ∪ (Qc ∩ Q)

∵ Qc ∩ Q = ϕ 

⇒ A = Pc ∪ ϕ 

⇒ A = Pc

Additional Information

ϕ is the empty set and it is also called the complement of the universal set. 

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