Let's construct the truth table for the given logical expression:

$(\neg p \rightarrow q) \land [(\neg q \leftrightarrow p) \rightarrow (p \lor \neg q)]$

We'll break down the expression into simpler components and then build the truth table.

Components to Evaluate:

1. $\neg p$
2. $\neg q$
3. $\neg p \rightarrow q$
4. $\neg q \leftrightarrow p$
5. $p \lor \neg q$
6. $(\neg q \leftrightarrow p) \rightarrow (p \lor \neg q)$
7. $(\neg p \rightarrow q) \land [(\neg q \leftrightarrow p) \rightarrow (p \lor \neg q)]$

Truth Table:

We'll consider all possible truth values for $p$ and $q$.

$p$	$q$	$\neg p$	$\neg q$	$\neg p \rightarrow q$	$\neg q \leftrightarrow p$	$p \lor \neg q$	$(\neg q \leftrightarrow p) \rightarrow (p \lor \neg q)$	$(\neg p \rightarrow q) \land [(\neg q \leftrightarrow p) \rightarrow (p \lor \neg q)]$
T	T	F	F	T	F	T	T	T
T	F	F	T	F	T	T	T	F
F	T	T	F	T	F	F	T	T
F	F	T	T	F	T	T	T	F

Conclusion:

• The expression evaluates to true (T) when both p and q are true or when p is false and q is true.
• The expression evaluates to false (F) in the other cases.

Would you like further details on how the calculations were done, or do you have any other questions?

Related Questions:

1. How is the implication ($\rightarrow$) operation defined in logical truth tables?
2. What does the biconditional ($\leftrightarrow$) operation signify in logical expressions?
3. How do De Morgan's laws apply to logical negations?
4. Can the given expression be simplified further using logical equivalences?
5. How would this expression behave in a situation where $p$ and $q$ represent real-world propositions?

Tip:

Always start constructing a truth table by breaking down the logical expression into its simplest components, which allows for easier and more accurate evaluations.