Answer to Question #212772 in Discrete Mathematics for dipayan

Question

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

Answer 1

$(p → r) ∨ (q → r)$

$\equiv (\neg p\lor r) \lor (\neg q\lor r) \ \ \ \ \ \ \$ (Implication)

$\equiv \neg p\lor r \lor \neg q\lor r \ \ \ \ \ \ \$

$\equiv (\neg p \lor \neg q)\lor r \ \ \ \ \ \ \$ (Distribution)

$\equiv \neg ( p\land q)\lor r$ (De Morgan's Law)

$\equiv ( p\land q) → r$ (Implication)

Hence proved ,

$(p → r) ∨ (q → r)$ and $( p\land q) → r$ are logically equivalent.