Question

(i) $p \supset (q \cdot r)$ (ii) $\sim(p \supset s)$
Taking (i) and (ii) as premises, which of the following can be deduced?

• A. $q$
• B. $r$
• C. $s$
• D. $\sim p$

Hint:

Always convert implications using Material Implication ($p \supset q \equiv \sim p \lor q$). It often simplifies proofs when combined with De Morgan's Theorem.

Answer 1

The Correct Option is A, B

From (ii) $\ \sim(p \supset s)$, apply Material Implication: $p \supset s \equiv \sim p \lor s$. Hence \[ \sim(p \supset s)=\sim(\sim p \lor s)\stackrel{\text{De Morgan}}{=} \sim\sim p \ \cdot\ \sim s = p \ \cdot\ \sim s. \] So we have $p$ and $\sim s$.
Using (i) and Modus Ponens with $p$: \[ p, p \supset (q \cdot r) \ \Rightarrow\ q \cdot r. \] By Simplification: \[ q \cdot r \ \Rightarrow\ q \text{and} r. \] Thus (A) $q$ and (B) $r$ are deducible; (C) $s$ and (D) $\sim p$ are not.