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.