If P, Q, R are subsets of the universal set U, then the value of following…
2008
If P, Q, R are subsets of the universal set U, then the value of following expression is
(P ∩ Q ∩ R) ∪ (Pc ∩ Q ∩ R) ∪ Qc ∪ Rc- A.
Qc U Rc
- B.
P U Qc U Rc
- C.
Pc U Qc U Rc
- D.
U
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Correct answer: D
Step 1: Combine the first two terms: (P ∩ Q ∩ R) ∪ (Pc ∩ Q ∩ R) = (P ∪ Pc) ∩ Q ∩ R = Q ∩ R.
Step 2: Replace these in the entire expression to get (Q ∩ R) ∪ Qc ∪ Rc.
Simplify Qc ∪ (Q ∩ R): Qc ∪ (Q ∩ R) = (Qc ∪ Q) ∩ (Qc ∪ R) = U ∩ (Qc ∪ R) = Qc ∪ R.
Now simplify (Qc ∪ R) ∪ Rc: (Qc ∪ R) ∪ Rc = Qc ∪ (R ∪ Rc) = Qc ∪ U = U.
Conclusion: The given expression simplifies to the universal set U.
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