**Question 5** (1 point)

*p → q*

*¬ (p* ∨ *q)*

*¬ q ↔ ¬ p*

Every proposition in the above list is True when

Question 5 options:

None of the above

Both *p* and *q* are True

*p* is True and *q* is False

*p* is False and *q* is True

Both *p* and *q* are False

To determine when all the propositions in the list are true, we need to evaluate each one based on different truth values of \( p \) and \( q \). The propositions given are:

1. \( p \rightarrow q \)
2. \( \neg(p \vee q) \)
3. \( \neg q \leftrightarrow \neg p \)

Let's go step-by-step for each proposition:

### Proposition 1: \( p \rightarrow q \) (Implication)
This means "if \( p \), then \( q \)." It's false only when \( p \) is true and \( q \) is false. In all other cases, it is true.

### Proposition 2: \( \neg(p \vee q) \) (Negation of OR)
This is equivalent to \( \neg p \wedge \neg q \) (De Morgan's Law). It means that both \( p \) and \( q \) must be false for the proposition to be true. 

### Proposition 3: \( \neg q \leftrightarrow \neg p \) (Biconditional)
This is true when \( \neg q \) and \( \neg p \) have the same truth values. In other words, it is true if both \( p \) and \( q \) are either true together or false together.

### Evaluating each case:

| \( p \)  | \( q \)  | \( p \rightarrow q \)  | \( \neg(p \vee q) \)  | \( \neg q \leftrightarrow \neg p \)  | All true? |
|:-------:|:-------:|:---------------------:|:--------------------:|:----------------------------:|:---------:|
|   T     |   T     |          T             |         F            |             T                |    No     |
|   T     |   F     |          F             |         F            |             F                |    No     |
|   F     |   T     |          T             |         F            |             F                |    No     |
|   F     |   F     |          T             |         T            |             T                |    **Yes**    |

### Conclusion:
All three propositions are true **only when both \( p \) and \( q \) are false**. 

Thus, the correct answer is:
- **Both \( p \) and \( q \) are False.**

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Would you like further details or have any questions? Here are five related questions for exploration:

1. How does the truth table for \( p \rightarrow q \) differ from \( p \leftrightarrow q \)?
2. Can you provide an example where \( p \vee q \) is false?
3. How does De Morgan’s law help simplify logical expressions?
4. What are the possible truth values of a biconditional statement \( p \leftrightarrow q \)?
5. How do you interpret negation in logic?

**Tip**: To solve complex logical propositions, truth tables can simplify the analysis of each possible case.

Every proposition in the following list is True when: p → q, ¬(p ∨ q), ¬q ↔ ¬p. Options: None of the above, Both p and q are True, p is True and q is False, p is False and q is True, Both p and q are False.

Explore a detailed solution to the logical problem involving propositions p → q, ¬(p ∨ q), and ¬q ↔ ¬p. Learn how to evaluate truth tables, use De Morgan’s Law, and determine when all propositions are true. Suitable for students in Grades 10-12.

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