The image contains four logical expressions involving propositions $p$, $q$, and $r$. Let's analyze each of them:

1. $\neg(p \land \neg q)$

   • This expression is the negation of $p \land \neg q$, which states that $p$ is true and $q$ is false. Negating it will result in $\neg p \lor q$ (by De Morgan's laws).

2. $p \lor \neg(r \land q)$

   • This is a disjunction where one term is $p$, and the other is the negation of the conjunction $r \land q$. Using De Morgan's laws, $\neg(r \land q)$ can be rewritten as $\neg r \lor \neg q$. Therefore, this expression simplifies to $p \lor (\neg r \lor \neg q)$.

3. $\neg((\neg q \lor p) \land \neg r)$

   • This is the negation of a conjunction. By De Morgan's laws, it will become $\neg(\neg q \lor p) \lor r$. The term $\neg(\neg q \lor p)$ is equivalent to $q \land \neg p$. Therefore, the entire expression becomes $(q \land \neg p) \lor r$.

4. $(q \lor r) \land (\neg p \lor \neg r)$

   • This is a conjunction of two disjunctions. It states that both $q \lor r$ and $\neg p \lor \neg r$ must be true simultaneously.

If you'd like further details, let me know.

Here are some related questions:

1. How does De Morgan's law apply in logic simplifications?
2. What is the meaning of the conjunction and disjunction in propositional logic?
3. How can truth tables be used to verify logical expressions?
4. Can logical equivalence be used to simplify any of the given expressions further?
5. What are common logical fallacies or mistakes made during propositional logic transformations?

Tip: Always verify the simplifications of logical expressions using truth tables to confirm their accuracy.