Boolean Expression Analysis

We need to find the expressions equivalent to the given Boolean function:

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

Simplifying the Expression F

We simplify $F$ using algebraic manipulation:

1. Let $A = \bar{P} + Q$ and $B = \bar{P}Q$. The expression is $F = A \oplus B$.
2. Use the XOR definition: $X \oplus Y = X\bar{Y} + \bar{X}Y$.
3. Calculate the complements $\bar{A}$ and $\bar{B}$:
   • $\bar{A} = \overline{(\bar{P} + Q)} = \overline{\bar{P}}\overline{Q} = P\bar{Q}$ (Using De Morgan's Law).
   • $\bar{B} = \overline{(\bar{P}Q)} = \overline{\bar{P}} + \bar{Q} = P + \bar{Q}$ (Using De Morgan's Law).
4. Substitute back into the XOR formula: $F = (\bar{P} + Q)(P + \bar{Q}) + (P\bar{Q})(\bar{P}Q)$
5. Expand and simplify the terms:
   • First term: $(\bar{P} + Q)(P + \bar{Q}) = \bar{P}P + \bar{P}\bar{Q} + QP + Q\bar{Q} = 0 + \bar{P}\bar{Q} + PQ + 0 = PQ + \bar{P}\bar{Q}$.
   • Second term: $(P\bar{Q})(\bar{P}Q) = P\bar{P}\bar{Q}Q = 0$.
6. Combine the results: $F = (PQ + \bar{P}\bar{Q}) + 0 = PQ + \bar{P}\bar{Q}$.

The simplified expression for $F$ is $PQ + \bar{P}\bar{Q}$, which is the expression for XNOR.

Evaluating Equivalent Options

We check which options are equivalent to $PQ + \bar{P}\bar{Q}$:

• Option 1: $\overline{P \oplus Q}$

  This is the definition of XNOR, equivalent to $PQ + \bar{P}\bar{Q}$. It matches $F$.

• Option 2: $P \oplus Q$

  This is XOR, equivalent to $PQ + \bar{P}Q$. It does not match $F$.

• Option 3: $\bar{P} \oplus Q$

  Algebraically: $\bar{P} \oplus Q = (\bar{P})\overline{Q} + \overline{(\bar{P})}Q = \bar{P}\bar{Q} + PQ$. This matches $F$.

• Option 4: $\bar{P} \oplus \bar{Q}$

  Algebraically: $\bar{P} \oplus \bar{Q} = (\bar{P})\overline{(\bar{Q})} + \overline{(\bar{P})}\bar{Q} = \bar{P}Q + P\bar{Q}$. It does not match $F$.

Verification with Truth Table

A truth table confirms the equivalences:

The output columns for $F$, Option 1, and Option 3 are identical, confirming their equivalence.

Final Answer Determination

The expressions equivalent to $F(P, Q)$ are Option 1 ($\overline{P \oplus Q}$) and Option 3 ($\bar{P} \oplus Q$).