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.

Explore the logical equivalence between ¬(p → q) and (p ∧ ¬q) in propositional logic. This problem illustrates the application of De Morgan’s law and discusses how biconditional statements work. Suitable for advanced high school or undergraduate students.

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