Question

If P, Q, R are subsets of the universal set U, then (P $\cap$ Q $\cap$ R) $\cup$ ($P^c$ $\cap$ Q $\cap$ R) $\cup$ $Q^c$ $\cup$ $R^c$ is

• A. $Q^c$ $\cup$ $R^c$
• B. P $\cup$ $Q^c$ $\cup$ $R^c$
• C. $P^c$ $\cup$ $Q^c$ $\cup$ $R^c$
• D. U

Answer 1

The correct option is D U

The given set theory expression can be convert in to equivalent boolean algebra expression follows:
(p $\cap$ q $\cap$ r) $\cup$ ($p^c$ $\cap$ q $\cap$ r) $\cup$ $q^c$ $\cup$ $r^c$
= pqr + p'qr + q' + r'
= qr + (p + P') + q' + r'
= qr + q' + r'
=(q+q').(r+q')+r'
= r + q' + r'
= r + r' + q'
= 1 + q'
= 1 = U