Question

Consider the following Boolean expression of a function \( F \):

\[ F(P,Q) = (\bar{P} + Q) \oplus (\bar{P}Q) \]

Which of the following expressions is/are equivalent to \( F \)?

• A. \( \overline{P \oplus Q} \)
• B. \( P \oplus Q \)
• C. \( \bar{P} \oplus Q \)
• D. \( \bar{P} \oplus \bar{Q} \)

Hint:

Simplify using the identity A XOR B = A AND (NOT B) when B always implies A, then compare with a truth table.

Answer 1

The Correct Option is A, C

Method: Simplify \( F \) algebraically using Boolean identities instead of a truth table.
Let \( A = \bar{P}+Q \) and \( B = \bar{P}Q \). Notice \( B \) implies \( A \): whenever \( \bar{P}Q=1 \), \( \bar{P}=1 \), so \( \bar{P}+Q \) is automatically 1 too, i.e. \( AB = B \).

For Boolean values where \( B \) implies \( A \), \( A \oplus B = A\bar{B} \), since XOR is 1 exactly when A and B differ, and the only way for that is \( A=1, B=0 \).

So \( F = A\bar{B} = (\bar{P}+Q)\cdot\overline{\bar{P}Q} \). By De Morgan's law, \( \overline{\bar{P}Q} = P+\bar{Q} \), so \( F = (\bar{P}+Q)(P+\bar{Q}) \).

Expand: \( F = \bar{P}P + \bar{P}\bar{Q} + QP + Q\bar{Q} = 0 + \bar{P}\bar{Q} + PQ + 0 = \bar{P}\bar{Q}+PQ \).

\( \bar{P}\bar{Q}+PQ \) is by definition the XNOR of P and Q, i.e. \( F = \overline{P \oplus Q} \), which is option (A).

Also, \( \bar{P} \oplus Q = \bar{P}\bar{Q} + PQ \) (XOR of \( \bar{P} \) and Q is true when they differ: \( \bar{P}=1,Q=0 \) giving \( \bar{P}\bar{Q} \), or \( \bar{P}=0,Q=1 \) giving PQ), which is identical to the simplified F. So option (C) also matches.

Options (B) and (D) both reduce to \( P\bar{Q}+\bar{P}Q \), the complement of F, so neither matches.

\[ \boxed{\text{Options (A) and (C)}} \]