Question

Consider the following Boolean expression of a function F :
𝐹(𝑃, 𝑄) = (𝑃̅ + 𝑄) ⊕(𝑃̅𝑄)
Which of the following expressions is/are equivalent to F ?

• A. 𝑃⊕𝑄 ̅̅̅̅̅̅̅̅
• B. 𝑃⊕𝑄
• C. 𝑃̅ ⊕𝑄
• D. 𝑃̅ ⊕𝑄̅

Hint:

Build the truth table of F for all four combinations of P and Q, then compare it row by row against each option; F turns out to be the XNOR of P and Q.

Answer 1

The Correct Option is A, C

Let's simplify F(P, Q) = \((\overline{P} + Q) \oplus (\overline{P}Q)\) algebraically instead of building a truth table.

Step 1: Let \(A = \overline{P}+Q\) and \(B = \overline{P}Q\).

Notice that whenever \(B\) is true, \(A\) is automatically true too, since \(\overline{P}Q\) being true forces \(\overline{P}+Q\) to be true. So \(B\) implies \(A\), which means \(AB = B\) and \(A+B = A\).

Step 2: Use the identity \(A \oplus B = (A+B)\cdot\overline{AB}\).

Substituting \(A+B = A\) and \(AB = B\), we get \(F = A \cdot \overline{B} = (\overline{P}+Q)\cdot\overline{\overline{P}Q} = (\overline{P}+Q)(P+\overline{Q})\).

Step 3: Expand \((\overline{P}+Q)(P+\overline{Q})\).

\(= \overline{P}P + \overline{P}\overline{Q} + QP + Q\overline{Q} = 0 + \overline{P}\overline{Q} + PQ + 0 = \overline{P}\overline{Q} + PQ\), which is exactly the standard sum-of-products form of the XNOR function \(\overline{P \oplus Q}\).

Step 4: Match with the options.

Option (A) is literally \(\overline{P \oplus Q}\), identical to our derived result, so A is correct. Option (C) is \(\overline{P} \oplus Q\), which equals 1 exactly when \(\overline{P}\) and \(Q\) differ, i.e. when P equals Q, giving the same expression \(\overline{P}\overline{Q}+PQ\), so C is also correct. Options B and D reduce to the plain XOR \(P\overline{Q}+\overline{P}Q\), the complement of our result, so they are wrong.

Hence the expressions equivalent to F are options A and C.