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Find the complement of the following expression and reduce it by using Boolean laws.
Find the complement of the following expression and reduce it by using Boolean laws.
$P \cdot (P + Q) \cdot Q \cdot (Q + R')$
Answer
Answer
AILet E = P.(P+Q).Q.(Q+R')
By absorption law, P.(P+Q) = P, so E = P.Q.(Q+R')
Again by absorption law, Q.(Q+R') = Q, so E = P.Q
Complement of E: E' = (P.Q)' = P' + Q' (De Morgan's law)
Verification by taking complement of the original expression directly:
E' = [P.(P+Q).Q.(Q+R')]'
= P' + (P+Q)' + Q' + (Q+R')' (De Morgan's law)
= P' + P'.Q' + Q' + Q'.R (De Morgan's law again)
= P' + Q' + Q'.R (absorption: P' + P'.Q' = P')
= P' + Q' (absorption: Q' + Q'.R = Q')
Reduced complement: E' = P' + Q'
From ISC 2024 Computer Science Paper 1, question 4(iii).