Final answer:
The correct next step in the proof by tableau method for determining if Z = [p v q] ⇒ p is a tautology is C. Tp Fq Fp Fq. This step examines the possibilities when '[p v q] ⇒ p' is assumed to be false and shows the branches that lead to a contradiction, hinting that the original proposition is indeed a tautology.
Step-by-step explanation:
The student is asking whether the logical statement Z = [p v q] ⇒ p is a tautology, and which would be the next step in a proof by tableau for showing this. In a tableau method, we aim to show that the negation of the proposition leads to a contradiction. Given that we start with FZ = F[p v q] ⇒ p, the next step would involve breaking down the logical formula based on the rules that apply to the logical connectors involved here, which are 'or' (v) and 'implies' (⇒).
The correct next step would be C. Tp Fq Fp Fq. This step adheres to the rules of logical disjunction and implication in a proof by tableau. If '[p v q] ⇒ p' were false, 'p v q' would have to be true and 'p' would have to be false at the same time. It is then necessary to break down the 'or' statement into two branches, one where 'p' is true (Tp) and one where 'q' is true (Fq). Since we already have that 'p' is false in the original assumption, we add Fp to both branches. This leads to an evident contradiction, as we cannot have both Tp and Fp in the same branch (branches must be consistent). The contradiction in the tableau thus hints at the fact that the negation of the original proposition cannot be true, which suggests that the original proposition is indeed a tautology.