The problem asks to show that $\neg p \rightarrow (q \rightarrow r)$ and $q \rightarrow (p \vee r)$ are logically equivalent. Let's go step by step through the process of proving their equivalence using logical transformations.

Given Expression

$\neg p \rightarrow (q \rightarrow r)$ This is the initial statement that needs to be shown as logically equivalent to: $q \rightarrow (p \vee r)$

Steps and Justifications

Here are the ordered steps with justifications to transform $\neg p \rightarrow (q \rightarrow r)$ into $q \rightarrow (p \vee r)$:

1. Start with $\neg p \rightarrow (q \rightarrow r)$.

2. Apply the implication rule ($a \rightarrow b = \neg a \vee b$) to both implications: $\neg p \rightarrow (q \rightarrow r) \equiv p \vee (\neg q \vee r)$ This step involves converting both implications into disjunctions.

3. The expression simplifies to: $p \vee \neg q \vee r$

4. Rearrange the terms: $\neg q \vee (p \vee r)$ This uses the associative and commutative properties of disjunction.

5. Finally, recognize that the expression now matches the form $q \rightarrow (p \vee r)$, since $q \rightarrow (p \vee r)$ is logically equivalent to $\neg q \vee (p \vee r)$.

Correct Order of Drag-and-Drop Boxes

The correct sequence for the drag-and-drop boxes based on this logical derivation is:

1. $\neg p \rightarrow (q \rightarrow r)$
2. $= p \vee (q \rightarrow r)$
3. $= p \vee (\neg q \vee r)$
4. $= p \vee \neg q \vee r$
5. $= \neg q \vee (p \vee r)$
6. $= q \rightarrow (p \vee r)$

This shows the logical equivalence step-by-step.

Would you like more details about any step? Here are five related questions to deepen your understanding:

1. What are the logical equivalence rules used in proving statements?
2. How does the implication rule $a \rightarrow b = \neg a \vee b$ work?
3. Why are the associative and commutative properties important in logical expressions?
4. How can you identify if two logical expressions are equivalent?
5. What other methods can be used to prove logical equivalences?

Tip: When proving logical equivalences, always check if you can apply basic logical laws like De Morgan’s laws, distributive, commutative, and associative properties for simplification.