In propositional logic if \( (P → Q) ∧ (R → S)\) and \((P ∨ R)\) are two…
2017
In propositional logic if \( (P → Q) ∧ (R → S)\) and \((P ∨ R)\) are two premises such that
\(\begin{array}{c} (P \to Q) \wedge (R \to S) \\ P \vee R \\ \hline Y \\ \hline \end{array}\)
\(Y\) is the premise :
- A.
\(𝑃∨𝑅 \) - B.
\(𝑃∨𝑆 \) - C.
\(𝑄∨𝑅 \) - D.
\(𝑄∨𝑆\)
Attempted by 123 students.
Show answer & explanation
Correct answer: D
Solution: Use the premises to derive the conclusion.
From (P → Q) ∧ (R → S) extract P → Q and R → S by conjunction elimination.
We are also given the disjunction P ∨ R.
Apply the constructive dilemma: from P → Q, R → S, and P ∨ R we can infer Q ∨ S.
Therefore the valid derived conclusion is Q ∨ S.
Key idea: using the implications for each disjunct yields the disjunction of their consequents (constructive dilemma).
A video solution is available for this question — log in and enroll to watch it.