https://file.aiquickdraw.com/mathbot/file/17259147434468q03b44n.jpg

Resolve the logical proposition in this image

The logical proposition given in the image is:

\[
((p \rightarrow q) \rightarrow r) \leftrightarrow (p \land (\neg r \rightarrow \neg q))
\]

Let's resolve and simplify this logical proposition step by step.

### Step 1: Simplify \( \neg r \rightarrow \neg q \)
Using the equivalence \( \neg r \rightarrow \neg q \equiv q \rightarrow r \), we can rewrite the right side of the equivalence:

\[
p \land (\neg r \rightarrow \neg q) \equiv p \land (q \rightarrow r)
\]

### Step 2: Rewrite the whole expression
Now substitute the simplified expression into the equivalence:

\[
((p \rightarrow q) \rightarrow r) \leftrightarrow (p \land (q \rightarrow r))
\]

### Step 3: Break down \( p \rightarrow q \)
Recall that \( p \rightarrow q \equiv \neg p \lor q \). Substitute this into the left side:

\[
(\neg p \lor q) \rightarrow r
\]

This can be simplified using \( (\neg p \lor q) \rightarrow r \equiv \neg (\neg p \lor q) \lor r \equiv (p \land \neg q) \lor r \).

### Step 4: Compare the two sides of the equivalence
The left side is \( ((p \land \neg q) \lor r) \), and the right side is \( p \land (q \rightarrow r) \).

So the final simplified expression is:

\[
((p \land \neg q) \lor r) \leftrightarrow (p \land (q \rightarrow r))
\]

This expression shows the logical equivalence of the original proposition.

Would you like a more detailed breakdown or have any questions?

### Follow-up Questions:
1. How do you simplify logical expressions using truth tables?
2. What are common equivalences in logical propositions?
3. How does the equivalence \( \neg r \rightarrow \neg q \equiv q \rightarrow r \) work?
4. How can logical equivalences be used in computer programming?
5. What is the importance of simplifying logical propositions?

**Tip:** Always double-check logical equivalences using truth tables to ensure accuracy.

Step-by-step resolution and simplification of the logical proposition ((p → q) → r) ↔ p ∧ (¬r → ¬q), including key logical equivalences and simplifications. Suitable for undergraduate-level students studying logic or discrete mathematics.

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