Let's simplify and analyze the given logical expression step by step. The expression is:

¬(¬p ∨ r) → (q → p), q → (p ∨ r)

Step 1: Break down the components

Part 1: ¬(¬p ∨ r) → (q → p)

1. ¬(¬p ∨ r): Using De Morgan's law:
   ¬(¬p ∨ r) ≡ ¬¬p ∧ ¬r ≡ p ∧ ¬r.

2. q → p: This can be rewritten as ¬q ∨ p (implication equivalence).

So, the first part becomes: (p ∧ ¬r) → (¬q ∨ p).

Using implication equivalence again: (p ∧ ¬r) → (¬q ∨ p) ≡ ¬(p ∧ ¬r) ∨ (¬q ∨ p).

Simplify ¬(p ∧ ¬r):

¬(p ∧ ¬r) ≡ ¬p ∨ r.

Thus, the first part simplifies to: (¬p ∨ r) ∨ (¬q ∨ p).

Reorganize terms: ¬q ∨ p ∨ r.

Part 2: q → (p ∨ r)

Rewrite q → (p ∨ r) as: ¬q ∨ (p ∨ r).

Step 2: Combine the two parts

The full expression is: (¬q ∨ p ∨ r) ∧ (¬q ∨ (p ∨ r)).

Step 3: Simplify further

The second part, ¬q ∨ (p ∨ r), already overlaps with the first part, so the entire expression can be written as: ¬q ∨ p ∨ r.

Final Simplified Expression:

¬q ∨ p ∨ r

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Would you like further explanation, examples, or details? Here are some related questions to deepen your understanding:

1. What are De Morgan's laws, and how do they help in logical expressions?
2. How does implication equivalence (p → q ≡ ¬p ∨ q) work in logical simplifications?
3. How do you convert a compound logical expression into conjunctive or disjunctive normal form?
4. What is the difference between logical validity and satisfiability?
5. How can truth tables be used to verify logical equivalences?

Tip: Always apply De Morgan's laws and implication equivalences first when simplifying logical expressions. It makes subsequent steps easier.