Here is an unfinished proof of the following sequent:
𝑃
∧
(
𝑄
∨
𝑅
)
⊢
(
𝑃
∧
𝑄
)
∨
(
𝑃
∧
𝑅
)

Which one of the following would become a correct line (10) that makes the proof complete?

Question

Here is an unfinished proof of the following sequent: 
𝑃
∧
(
𝑄
∨
𝑅
)
⊢
(
𝑃
∧
𝑄
)
∨
(
𝑃
∧
𝑅
)

Which one of the following would become a correct line (10) that makes the proof complete?

BlackTom AI · Accepted Answer

The question asks which line would correctly complete the unfinished proof of the sequent P ∧ (Q ∨ R) ⊢ (P ∧ Q) ∨ (P ∧ R).

First, recall a standard logical equivalence: distribution of ∧ over ∨ gives P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R). In a formal proof, this often corresponds to introducing a disjunction of two conjunctions via a distributive step.

Option 1 analysis:
- The line labeled as (10) displays: (P ∧ Q) ∨ (P ∧ R).
- This is exactly the distributive form we expect from P ∧ (Q ∨ R). Placing this as line (10) would complete the intended derivation to reach the goal (P ∧ Q) ∨ (P ∧ R).
- Therefore, option 1 is appropriate because it represents the correct distributive consequence of the given premise, aligning with the standard logical transformation.

About the other options:
- The problem statement provides only one option string for the answer, so we cannot evaluate alternative lines that might appear as options 2, 3, etc. In a full proof, each alternate line would need to be checked for validity within the proof system (e.g., whether it follows from previous lines by an allowed rule) and whether it advances toward the target sequent.
- If any other option claimed to be line (10) but was not the correct distributive consequence, it would fail to match the necessary equivalence or would not logically follow from the prior steps.

In summary, the line (10) with (P ∧ Q) ∨ (P ∧ R) is the correct distributive consequence needed to bridge from P ∧ (Q ∨ R) to the desired disjunction of conjunctions, provided it is properly justified by a distributive step. Since only one option is provided here, we can identify it as the correct form for the completion given the usual proof structure.

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

Here is an unfinished proof of the following sequent: 𝑃 ∧ ( 𝑄 ∨ 𝑅 ) ⊢ ( 𝑃 ∧ 𝑄 ) ∨ ( 𝑃 ∧ 𝑅 ) Which one of the following would become a correct line (10) that makes the proof complete?

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