To create a truth table for the given logical expression:

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

I'll first break down the expression into smaller components, compute each part, and then use these to form the complete truth table. Let's proceed step by step.

Step 1: Identify the variables

The variables involved are $p$, $q$, and $r$. Since we have three variables, there will be $2^3 = 8$ possible combinations of truth values.

Step 2: Compute the truth values of each component

1. $\neg r$ (the negation of $r$)
2. $p \leftrightarrow \neg r$ (biconditional between $p$ and $\neg r$)
3. $\neg q$ (the negation of $q$)
4. $\neg q \rightarrow r$ (implication from $\neg q$ to $r$)
5. $\neg [(\neg q \rightarrow r) \land \neg p]$ (negation of the conjunction of the implication and $\neg p$)
6. Finally, $(p \leftrightarrow \neg r) \lor \neg [(\neg q \rightarrow r) \land \neg p]$ (the disjunction of the biconditional and the negation).

Step 3: Construct the truth table

I'll generate the truth table based on these components.

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

Explanation:

• $p, q, r$: The basic truth values.
• $\neg r$: Negation of $r$.
• $p \leftrightarrow \neg r$: True when $p$ and $\neg r$ have the same truth value.
• $\neg q$: Negation of $q$.
• $\neg q \rightarrow r$: True when either $\neg q$ is false or $r$ is true.
• $(\neg q \rightarrow r) \land \neg p$: True when both $(\neg q \rightarrow r)$ and $\neg p$ are true.
• $\neg[(\neg q \rightarrow r) \land \neg p]$: The negation of the above conjunction.
• Final column: The result of the disjunction in the original expression.

This truth table represents all possible outcomes for the given logical expression.

Would you like further details, or do you have any questions?

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Here are five follow-up questions that build on this information:

1. How does the biconditional operator $\leftrightarrow$ differ from the implication $\rightarrow$?
2. What effect does the negation operator have on logical expressions?
3. How would the truth table change if $p$ were replaced by $q$ in the original expression?
4. Can you identify any tautologies or contradictions in the given expression?
5. How can we simplify the given logical expression using logical equivalences?

Tip: When constructing truth tables, it's helpful to break down complex expressions into smaller components to make the process more manageable.