Let's break down the given logical expression:

[~r ∨ (r → ~q)] ∧ (~q → ~r)

To construct the truth table, we'll need to compute the following step by step:

1. Negation of $r$, denoted as ~r.
2. Implication $r → ~q$.
3. Disjunction of ~r and (r → ~q).
4. Implication $~q → ~r$.
5. The conjunction of [~r ∨ (r → ~q)] and (~q → ~r).

We'll use three variables: $r$, $q$, and their negations, leading to 8 rows in the truth table (since $2^3 = 8$).

Step-by-Step Table Construction:

$r$	$q$	$~r$	$~q$	$r → ~q$	$~r ∨ (r → ~q)$	$~q → ~r$	$[~r ∨ (r → ~q)] ∧ (~q → ~r)$
T	T	F	F	F	F	T	F
T	F	F	T	T	T	F	F
F	T	T	F	T	T	T	T
F	F	T	T	T	T	T	T
T	T	F	F	F	F	T	F
T	F	F	T	T	T	F	F
F	T	T	F	T	T	T	T
F	F	T	T	T	T	T	T

Final Truth Table Explanation:

• ~r is the negation of $r$.
• ~q is the negation of $q$.
• $r → ~q$ represents the implication that if $r$ is true, then $q$ must be false (i.e., $~q$ must be true).
• ~r ∨ (r → ~q) is the disjunction (OR) of ~r and $r → ~q$.
• ~q → ~r is the implication that if $~q$ is true, then $~r$ must also be true.
• [~r ∨ (r → ~q)] ∧ (~q → ~r) is the final conjunction (AND) of both parts.

Now, based on this truth table, you can evaluate any options or check your answers.

Do you have any follow-up questions, or would you like further details?

Related questions:

1. What is the definition of disjunction in logic?
2. How is the implication rule applied in a truth table?
3. How does negation affect a logical statement's truth value?
4. How do we interpret a conjunction in propositional logic?
5. What is the significance of truth tables in logic?

Tip:

Always check the truth values of both components of a conjunction; both must be true for the overall expression to be true.