We are tasked with simplifying the expression $(P \Rightarrow Q) \Rightarrow (\neg Q \Rightarrow \neg P).$

Step-by-step simplification:

1. Recall implications:

   • The implication $P \Rightarrow Q$ is logically equivalent to $\neg P \lor Q$.
   • The implication $\neg Q \Rightarrow \neg P$ is equivalent to $P \lor \neg Q$.

   So, we rewrite the expression: $(\neg P \lor Q) \Rightarrow (P \lor \neg Q)$

2. Expand the outer implication:

   • The implication $A \Rightarrow B$ is equivalent to $\neg A \lor B$. Applying this to our current expression: $\neg (\neg P \lor Q) \lor (P \lor \neg Q)$

3. Simplify the negation:

   • Use De Morgan's law to simplify $\neg (\neg P \lor Q)$: $\neg (\neg P \lor Q) = P \land \neg Q$ So the expression becomes: $(P \land \neg Q) \lor (P \lor \neg Q)$

4. Distribute and simplify: Now, simplify by combining terms:

   • The expression $(P \land \neg Q) \lor P$ simplifies to just $P$.
   • The remaining term $\lor \neg Q$ is already present.

   So the final simplified form is: $P \lor \neg Q$

Thus, the simplified expression is: $P \lor \neg Q$

Would you like further details on any of these steps?

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Here are 5 related questions to deepen your understanding:

1. What is De Morgan's Law, and how does it apply in logic?
2. How do you convert logical implications to disjunctions?
3. Can you simplify $(P \Rightarrow Q) \land (\neg Q \Rightarrow \neg P)$?
4. How does $P \lor \neg Q$ relate to the contrapositive of $P \Rightarrow Q$?
5. How do tautologies and contradictions simplify complex logical expressions?

Tip: Always remember that an implication $A \Rightarrow B$ is equivalent to $\neg A \lor B$, which can greatly help in simplifications.