Consider the following truth table for a wff
𝑊
formed using propositional variables
𝑃
,
𝑄
and
𝑅
:

Which one of the following wffs would be correct for
𝑊
?

Question

Consider the following truth table for a wff 
𝑊
formed using propositional variables 
𝑃
 , 
𝑄
 and 
𝑅
:

Which one of the following wffs would be correct for 
𝑊
?

BlackTom AI · Accepted Answer

The question asks us to select a wff that would be correct for W, given a truth table involving P, Q, and R.

Option structure: The single provided option is (
P ∨ Q
) ⇒ (
Q ∨ R
).

Analysis of this form involves understanding how an implication interacts with disjunctions:
- The antecedent is P ∨ Q, which is true exactly when at least one of P or Q is true.
- The consequent is Q ∨ R, which is true when at least one of Q or R is true.
- An implication A ⇒ B is false only when A is true and B is false; in all other cases it is true.

Applying this to (
P ∨ Q
) ⇒ (
Q ∨ R
), we would check the following scenarios:
- If P ∨ Q is false (i.e., both P and Q are false), the implication is automatically true regardless of R, because a false antecedent makes the whole statement true.
- If P ∨ Q is true, then at least one of P or Q is true; for the implication to hold, the consequent Q ∨ R must also be true, meaning either Q is true or R is true (or both).
- The only potential for a false implication occurs when P ∨ Q is true while Q ∨ R is false. That would require P ∨ Q to be true with both Q and R false, which is impossible because if Q is false and R is false, Q ∨ R is false, but P ∨ Q would be true only if P is true. This yields a case where P is true, Q is false, R is false, making A true and B false, thus A ⇒ B false. Therefore, the wff would distinguish that particular assignment.

Since the question provides only this option as a candidate, we can discuss its alignment with typical logical relations:
- The expression captures a dependency: the truth of the left side (P ∨ Q) imposes a requirement on the right side (Q ∨ R). In many problem settings, this kind of structure is used to encode a relation among P, Q, and R that the table would reflect.
- Other potential wffs would need to be presented for comparison, but with only this option available, the analysis centers on the implications of the disjunctions involved.

Note: If the truth table for W marks certain input combinations as yielding true or false, this option would be judged accordingly by checking each row: does the implication hold exactly on the rows where the table indicates W is true? Since we do not have the explicit table values here, we rely on the standard semantics of disjunctions and implication to understand how this wff behaves in general terms.

In summary, this option encodes a reasonable dependency between the antecedent P ∨ Q and the consequent Q ∨ R, and its truth behavior follows the standard rules for implication and disjunction. The absence of additional options limits a direct comparative verdict, but the logical form is coherent for representing a relation among P, Q, and R.

MATH3066 (ND) 2025 Mid-Semester Test Practice Set C

Consider the following truth table for a wff 𝑊 formed using propositional variables 𝑃 , 𝑄 and 𝑅 : Which one of the following wffs would be correct for 𝑊 ?

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