Let’s address the problems in the image:

23. Construct the truth table for $(p \leftrightarrow q) \oplus \neg(p \leftrightarrow q)$:

This involves:

1. $p \leftrightarrow q$: Logical equivalence between $p$ and $q$ (True if both are the same).
2. $\neg(p \leftrightarrow q)$: Negation of the equivalence.
3. $(p \leftrightarrow q) \oplus \neg(p \leftrightarrow q)$: Exclusive OR (True if one operand is true and the other is false).

The truth table:

The result is always true.

24. Show that $\neg \forall x (P(x) \to Q(x))$ and $\exists x (P(x) \land \neg Q(x))$ are logically equivalent:

Proof:

1. The expression $P(x) \to Q(x)$ is equivalent to $\neg P(x) \lor Q(x)$ (definition of implication).

2. $\neg \forall x (P(x) \to Q(x))$:

   • This means there exists an $x$ such that $\neg (P(x) \to Q(x))$.

   • Substituting the equivalence $P(x) \to Q(x)$, we have: $\neg (P(x) \to Q(x)) = \neg (\neg P(x) \lor Q(x)) = P(x) \land \neg Q(x).$

   • Therefore, $\neg \forall x (P(x) \to Q(x)) \equiv \exists x (P(x) \land \neg Q(x))$.

This completes the proof of equivalence.

Do you need further clarification or examples? Here are five questions to explore further:

1. Can you explain the operation $\oplus$ in Boolean algebra with examples?
2. How can we interpret the truth table result of $(p \leftrightarrow q) \oplus \neg(p \leftrightarrow q)$?
3. What are some other logical equivalences similar to $\neg \forall x$ and $\exists x$?
4. How do these concepts apply in predicate logic in computer science or mathematics?
5. Can you derive $P(x) \land \neg Q(x)$ directly using laws of logic?

Tip: Always simplify logical expressions step-by-step to ensure clarity and accuracy.