𝑝 and 𝑞 are propositions.
i. Construct a truth table to show the truth values of ~𝑝 → 𝑞 and 𝑝˅𝑞
ii. State, with a reason, if ~𝑝 → 𝑞 and 𝑝˅𝑞 are logically equivalent. 

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.

𝑝 and 𝑞 are propositions.
i. Construct a truth table to show the truth values of ~𝑝 → 𝑞 and 𝑝˅𝑞
ii. State, with a reason, if ~𝑝 → 𝑞 and 𝑝˅𝑞 are logically equivalent.

This detailed explanation constructs a truth table for ~p → q and p˅q and checks their logical equivalence. Learn about negation, implication, disjunction, and how to use truth tables to verify logical equivalence. Suitable for students in Grades 9-12.

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