# DISCRETE MATH: Use rules of inference to show that...

by VinnyCee
Tags: discrete, inference, math, rules
 P: 492 1. The problem statement, all variables and given/known data Use rules of inference to show that if $$\forall\,x\,(P(x)\,\vee\,Q(x))$$ and $$\forall\,x\,((\neg\,P(x)\,\wedge\,Q(x))\,\longrightarrow\,R(x))$$ are true, then $$\forall\,x\,(\neg\,R(x)\,\longrightarrow\,P(x))$$ is true. 2. Relevant equations Universal instantiation, Disjunctive syllogism, Conjunction. 3. The attempt at a solution 1) $$\forall\,x\,(P(x)\,\vee\,Q(x))$$ Premise 2) $$P(a)\,\vee\,Q(a)$$ Universal instantiation of (1) 3) $$\neg\,P(a)$$ Disjunctive syllogism of (2) 4) $$\forall\,x\,((\neg\,P(x)\,\wedge\,Q(x))\,\longrightarrow\,R(x))$$ Premise 5) $$(\neg\,P(a)\,\wedge\,Q(a))\,\longrightarrow\,R(a)$$ Universal instantiation of (4) 6) $$R(a)$$ Modus Ponens of (5) Here I am stuck, any suggestions?

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