Answer to Question #286384 in Discrete Mathematics for May

Question

prove a --> ( b V c ) using contradiction method and combination of inference rules and equivalence laws from these premises : 1. a --> ( d V b ) 2. d --> c

Answer 1

Let us prove using the method by contradiction.

Suppose that the premises $a \to ( d \lor b )$ and $d \to c$ are true, but the conclusion $a \to( b \lor c )$ is false.

The definition of implication implies $a$ is true and $b\lor c$ is false, and hence definition of disjunction implies $b$ is false and $c$ is false. Since $d \to c$ is true and $c$ is false, we conclude that $d$ is false. Consequently, $d\lor b$ is false, and thus $a \to ( d \lor b )$ is false. This contradiction proves the premises $a \to ( d \lor b )$ and $d \to c$ imply the conclusion $a \to( b \lor c ).$