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

Step 1: Set up the truth table for \( F(P,Q) = (\bar{P}+Q) \oplus (\bar{P}Q) \) by evaluating all four input combinations.
For \( P=0, Q=0 \): \( \bar{P}=1 \), \( \bar{P}+Q = 1 \), \( \bar{P}Q = 0 \), so \( F = 1 \oplus 0 = 1 \).
For \( P=0, Q=1 \): \( \bar{P}=1 \), \( \bar{P}+Q = 1 \), \( \bar{P}Q = 1 \), so \( F = 1 \oplus 1 = 0 \).
For \( P=1, Q=0 \): \( \bar{P}=0 \), \( \bar{P}+Q = 0 \), \( \bar{P}Q = 0 \), so \( F = 0 \oplus 0 = 0 \).
For \( P=1, Q=1 \): \( \bar{P}=0 \), \( \bar{P}+Q = 1 \), \( \bar{P}Q = 0 \), so \( F = 1 \oplus 0 = 1 \).

Step 2: Collect the truth table of \( F \): (P,Q,F) = (0,0,1), (0,1,0), (1,0,0), (1,1,1). This is 1 whenever P and Q are equal, which is exactly the XNOR function, i.e. \( F = \overline{P \oplus Q} \).

Step 3: Verify option (A), \( \overline{P \oplus Q} \).
\( P \oplus Q \) is 0,1,1,0 for the four rows, so its complement is 1,0,0,1, matching \( F \) at every row. (A) is equivalent to \( F \).

Step 4: Verify option (B), \( P \oplus Q \).
This gives 0,1,1,0, the exact opposite of \( F \) (1,0,0,1). (B) is NOT equivalent to \( F \).

Step 5: Verify option (C), \( \bar{P} \oplus Q \).
P=0,Q=0: \( \bar{P}=1 \), \( 1 \oplus 0 = 1 \), matches F=1.
P=0,Q=1: \( \bar{P}=1 \), \( 1 \oplus 1 = 0 \), matches F=0.
P=1,Q=0: \( \bar{P}=0 \), \( 0 \oplus 0 = 0 \), matches F=0.
P=1,Q=1: \( \bar{P}=0 \), \( 0 \oplus 1 = 1 \), matches F=1.
All four rows match, so (C) is equivalent to \( F \).

Step 6: Verify option (D), \( \bar{P} \oplus \bar{Q} \).
P=0,Q=0: \( \bar{P}=1,\bar{Q}=1 \), \( 1 \oplus 1 = 0 \), but F=1 here, mismatch. (D) is NOT equivalent to \( F \); in fact \( \bar{P} \oplus \bar{Q} \) always equals \( P \oplus Q \), the same as (B).

Step 7: Conclusion.
The expressions equivalent to \( F \) are option (A) and option (C).

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