To construct a truth table for the logical expression:

$(q \lor r) \leftrightarrow \neg (p \land r)$

We will break it down step by step:

Step-by-Step Explanation:

1. p, q, r: These are the variables that will take all combinations of truth values.
2. $q \lor r$: This is the logical disjunction (OR) between $q$ and $r$. It will be true if either $q$ or $r$ is true.
3. $p \land r$: This is the logical conjunction (AND) between $p$ and $r$. It will be true only if both $p$ and $r$ are true.
4. $\neg (p \land r)$: This is the negation of $p \land r$. It will be true if $p \land r$ is false.
5. $(q \lor r) \leftrightarrow \neg (p \land r)$: This is the biconditional (↔), which is true if both sides are equal (either both true or both false).

Let's construct the truth table:

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

Explanation of Columns:

• The first three columns show the possible truth values for $p$, $q$, and $r$.
• The next column shows the result of the disjunction $q \lor r$.
• Then, we calculate $p \land r$ and its negation $\neg (p \land r)$.
• Finally, we evaluate the biconditional $(q \lor r) \leftrightarrow \neg (p \land r)$.

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Would you like me to explain any part in more detail or answer any other related questions?

Here are some related questions to deepen the understanding:

1. How do we evaluate the truth table for other logical operators like conjunction or negation?
2. What is the difference between a biconditional (↔) and a conditional (→) operator in logic?
3. How do you use truth tables to prove logical equivalencies?
4. Can a biconditional expression be false? Under what conditions?
5. What role does negation play in logical expressions, and how does it affect the truth values?

Tip: When constructing truth tables, always break down the expression into its simplest components first (like $q \lor r$ and $p \land r$) before combining them into more complex operators like biconditionals or conditionals.