Answer to Question #291551 in Discrete Mathematics for hilly

Question

Show that (p → q) ∧ (p → r) and p → (q ∧ r) are logically equivalent using

logical equivalence laws.

Answer 1

From the left hand side,

$(p\rightarrow q)\land(p\rightarrow r)\equiv(\neg p\lor q)\land(\neg p\lor r)$ (by reduction of $\rightarrow$ )

$\equiv\neg p\lor(q\land r)$ (by idempotence of $\lor$ )

$\equiv p\rightarrow(q\land r)$ (by reduction of $\rightarrow$ )

Therefore, $(p\rightarrow q)\land(p\rightarrow r)\space and \space p\rightarrow(q\land r) \\$ are logically equivalent.