Click and drag the steps in the correct order to show that \( \neg p \rightarrow(q \rightarrow \eta) \) and \( q \rightarrow(p \vee \eta) \) are logically equivalent. (Note: While proving. prove the equivalence from \( \neg p \rightarrow(q \rightarrow \eta) \) to \( q \rightarrow(p \vee \eta)) \)
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To demonstrate that \( \neg p \rightarrow (q \rightarrow \eta) \) and \( q \rightarrow (p \vee \eta) \) are logically equivalent, we can use a series of logical steps. Let’s organize those steps in order: 1. **Start with the first expression:** Assume \( \neg p \rightarrow (q \rightarrow \eta) \). 2. **Rewrite implications using logical equivalences:** The implication can be rewritten as \( \neg \neg p \vee (q \rightarrow \eta) \), which simplifies to \( p \vee (q \rightarrow \eta) \). 3. **Further break down the remaining implication:** The term \( q \rightarrow \eta \) can be rewritten as \( \neg q \vee \eta \). Hence now we have \( p \vee (\neg q \vee \eta) \). 4. **Apply Associative Law of disjunction:** Reorganize the disjunctions: \( (p \vee \neg q) \vee \eta \). 5. **Now express this in terms of the second expression:** Notice that \( q \rightarrow (p \vee \eta) \) is equivalent to \( \neg q \vee (p \vee \eta) \). 6. **Thus, the disjunctions are equivalent:** Since \( (p \vee \neg q) \vee \eta \) can assert \( \neg q \) leads to \( p \vee \eta \), this means we've shown \( \neg p \rightarrow (q \rightarrow \eta) \) leads to \( q \rightarrow (p \vee \eta) \). Therefore, the expression \( \neg p \rightarrow (q \rightarrow \eta) \) is logically equivalent to \( q \rightarrow (p \vee \eta) \).
