Let's break down the two compound propositions in the image and construct their truth tables.

(a) $(p \rightarrow q) \leftrightarrow (\neg q \rightarrow \neg p)$

• $p \rightarrow q$ is a conditional statement, which is true except when $p$ is true and $q$ is false.
• $\neg q \rightarrow \neg p$ is the contrapositive of $p \rightarrow q$, which is logically equivalent to $p \rightarrow q$. So, we expect the entire statement to always be true, but we will verify this with a truth table.

(b) $(p \oplus q) \land (p \oplus \neg q)$

• $p \oplus q$ is the XOR (exclusive or) operation, which is true when exactly one of $p$ or $q$ is true.
• $p \oplus \neg q$ is true when exactly one of $p$ or $\neg q$ is true.
• The conjunction $\land$ of these two expressions will be true only when both are true.

Now, I will construct the truth tables for both (a) and (b).

Let me calculate them.Here are the truth tables for the given compound propositions:

(a) $(p \rightarrow q) \leftrightarrow (\neg q \rightarrow \neg p)$

(b) $(p \oplus q) \land (p \oplus \neg q)$

Would you like more details on any part of these solutions?

5 Related Questions:

1. How do logical equivalences work in truth tables?
2. What is the difference between XOR ($\oplus$) and OR ($\lor$)?
3. Why is the contrapositive always logically equivalent to the original conditional?
4. Can you explain how conjunction ($\land$) affects the truth value of a compound proposition?
5. How do we interpret truth table results in logic circuits?

Tip: Always simplify logical expressions using known equivalences to verify complex propositions.