# DISCRETE MATH: Use rules of inference to show that

1. Jan 25, 2007

### VinnyCee

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?