The given logical expression is:

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

Let's break it down:

Left Side: $\neg(p \rightarrow q)$

The negation of $p \rightarrow q$ can be rewritten using logical equivalences:

• $p \rightarrow q$ is equivalent to $\neg p \lor q$.
• So, $\neg(p \rightarrow q)$ is equivalent to $\neg(\neg p \lor q)$.

Using De Morgan's law, we can simplify:

$\neg(\neg p \lor q) \equiv p \land \neg q$

Right Side: $p \land \neg q$

This is already in its simplest form.

Final Expression:

Thus, $\neg(p \rightarrow q)$ simplifies to $p \land \neg q$, and the given expression:

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

is always true, since both sides are logically equivalent.

Conclusion:

The logical biconditional (↔) holds because both sides represent the same logical statement.

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Would you like further details on this or have any specific questions?

Here are 5 questions to expand on the topic:

1. What are the truth tables for $p \rightarrow q$ and $\neg(p \rightarrow q)$?
2. How does De Morgan’s law apply in propositional logic?
3. What is the relationship between conjunction (∧) and disjunction (∨)?
4. Can you explain the conditions under which a biconditional (↔) holds true?
5. How does negating an implication affect its truth value?

Tip: Understanding logical equivalences helps simplify complex propositions easily.