Let's go through the steps to solve this problem.

i. Constructing a truth table for $\neg p \to q$ and $p \vee q$:

We have two propositions, $p$ and $q$, each of which can take the values True (T) or False (F). Let's first define the logical operations involved:

• Negation ($\neg p$): This is True when $p$ is False and False when $p$ is True.
• Implication ($\to$): For $\neg p \to q$, this is False only when the antecedent ($\neg p$) is True and the consequent ($q$) is False. In all other cases, it's True.
• Disjunction ($\vee$): For $p \vee q$, this is True if at least one of $p$ or $q$ is True, and False if both are False.

Now, let's construct the truth table.

$p$	$q$	$\neg p$	$\neg p \to q$	$p \vee q$
T	T	F	T	T
T	F	F	T	T
F	T	T	T	T
F	F	T	F	F

ii. Determining if $\neg p \to q$ and $p \vee q$ are logically equivalent:

Two propositions are logically equivalent if they have the same truth values for all possible truth values of $p$ and $q$.

From the truth table, we can observe that the truth values of $\neg p \to q$ and $p \vee q$ are the same for all combinations of $p$ and $q$. Therefore, we conclude that:

$\neg p \to q \quad \text{is logically equivalent to} \quad p \vee q$

Reason: Both propositions result in the same truth values for all possible combinations of truth values of $p$ and $q$, which confirms their logical equivalence.

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Would you like further details on any part of the explanation or have any questions?

Here are 5 related questions to deepen your understanding:

1. How would the truth table look for the proposition $p \to q$?
2. What is the truth value of $p \vee q$ when both $p$ and $q$ are False?
3. Can two propositions be logically equivalent if their truth values differ in only one case?
4. How does the truth table for $\neg (p \wedge q)$ compare to that of $p \vee q$?
5. What logical law can be used to justify the equivalence of $\neg p \to q$ and $p \vee q$?

Tip: When constructing truth tables, it's helpful to break complex expressions into smaller parts, evaluating each step independently before combining results.