https://file.aiquickdraw.com/mathbot/file/1726095608401tlrky3cw.jpg

The logical expression in the image is:

\[
\neg (q \Rightarrow \neg p) \Rightarrow (\neg p \lor q)
\]

Let’s break it down and simplify:

### Step 1: Rewrite the implications using logical equivalences.

The implication \( q \Rightarrow \neg p \) can be rewritten as:
\[
q \Rightarrow \neg p \equiv \neg q \lor \neg p
\]
So the expression becomes:
\[
\neg (\neg q \lor \neg p) \Rightarrow (\neg p \lor q)
\]

### Step 2: Apply De Morgan's law to the negation.

De Morgan's law tells us that:
\[
\neg (\neg q \lor \neg p) \equiv q \land p
\]
Now the expression is:
\[
(q \land p) \Rightarrow (\neg p \lor q)
\]

### Step 3: Analyze the implication.

The expression \( (q \land p) \Rightarrow (\neg p \lor q) \) holds true because:
- \( q \land p \) implies both \( q \) and \( p \) are true.
- The right side \( \neg p \lor q \) will be true because either \( q \) is true (which it is if \( q \land p \) holds) or \( \neg p \) is true (which doesn't need to hold if \( q \) is already true).

Thus, this expression is a **tautology** (always true).

Let me know if you'd like a further breakdown or explanation of any step!

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Here are 5 related questions that may help you deepen your understanding:

1. How does De Morgan's law work in logical expressions?
2. What other equivalent transformations are possible in propositional logic?
3. Can you provide an example where \( q \land p \) is false, and verify if the implication still holds?
4. What are the truth table results for this expression?
5. How does implication (\( \Rightarrow \)) relate to disjunction (\( \lor \)) in logic?

**Tip:** Always convert implications to disjunctions in order to simplify complex logical expressions.

Explore the step-by-step simplification of the logical expression ¬(q ⇒ ¬p) ⇒ (¬p ∨ q). This problem involves key concepts of propositional logic, including De Morgan's Law and implications, and provides an understanding of why the expression is a tautology.

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