Consider the following Boolean expression of a function F:

F(P,Q) = (¯P +Q)⊕(¯PQ)

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

The equivalent expressions to the given Boolean function $F$ are:
1. $P\odot Q$
2. $P\oplus \overline{Q}$
3. $\overline{P}\oplus Q$

Step-by-Step Derivation and Simplification:

The given Boolean expression is:
$$F(P,Q)=(\overline{P}+Q)\oplus (\overline{P}Q)$$

Recall the definition of the XOR (Exclusive OR) operation. For any two Boolean variables $A$ and $B$:
$$A\oplus B=\overline{A}B+A\overline{B}$$

Let us define:
$A=\overline{P}+Q$
$B=\overline{P}Q$

First, let us find the complements of $A$ and $B$:
1. Using De Morgan's Law for $\overline{A}$:
$$\overline{A}=\overline{(\overline{P}+Q)}=P\overline{Q}$$
2. Using De Morgan's Law for $\overline{B}$:
$$\overline{B}=\overline{(\overline{P}Q)}=P+\overline{Q}$$

Now, substitute these back into the XOR formula:
$$F=\overline{A}B+A\overline{B}$$

Let us evaluate each term separately:
• The first term $\overline{A}B$:
$$\overline{A}B=(P\overline{Q})(\overline{P}Q)=(P\overline{P})(Q\overline{Q})=0\cdot 0=0$$
• The second term $A\overline{B}$:
$$A\overline{B}=(\overline{P}+Q)(P+\overline{Q})$$
Distributing the multiplication:
$$A\overline{B}=\overline{P}P+\overline{P}\overline{Q}+QP+Q\overline{Q}$$
Since $\overline{P}P=0$ and $Q\overline{Q}=0$, this simplifies to:
$$A\overline{B}=PQ+\overline{P}\overline{Q}$$

Combining both terms:
$$F=0+PQ+\overline{P}\overline{Q}=PQ+\overline{P}\overline{Q}$$

The expression $PQ+\overline{P}\overline{Q}$ is the standard representation of the XNOR (Exclusive-NOR) operator:
$$F=P\odot Q$$

Equivalences of XNOR:
The XNOR gate represents the complement of the XOR gate:
$$P\odot Q=\overline{P\oplus Q}$$
Using XOR properties, complementing one of the inputs of an XOR gate complements the overall output, which makes it equivalent to XNOR:
$$\overline{P\oplus Q}=P\oplus \overline{Q}=\overline{P}\oplus Q$$

Thus, the equivalent expressions from the given choices are:
• $P\odot Q$
• $P\oplus \overline{Q}$
• $\overline{P}\oplus Q$