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

by VinnyCee
Tags: discrete, inference, math, rules
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Jan25-07, 01:35 PM
P: 492
1. The problem statement, all variables and given/known data

Use rules of inference to show that if [tex]\forall\,x\,(P(x)\,\vee\,Q(x))[/tex] and [tex]\forall\,x\,((\neg\,P(x)\,\wedge\,Q(x))\,\longrightarrow\,R(x))[/tex] are true, then [tex]\forall\,x\,(\neg\,R(x)\,\longrightarrow\,P(x))[/tex] is true.

2. Relevant equations

Universal instantiation, Disjunctive syllogism, Conjunction.

3. The attempt at a solution

1) [tex]\forall\,x\,(P(x)\,\vee\,Q(x))[/tex] Premise

2) [tex]P(a)\,\vee\,Q(a)[/tex] Universal instantiation of (1)

3) [tex]\neg\,P(a)[/tex] Disjunctive syllogism of (2)

4) [tex]\forall\,x\,((\neg\,P(x)\,\wedge\,Q(x))\,\longrightarrow\,R(x))[/tex] Premise

5) [tex](\neg\,P(a)\,\wedge\,Q(a))\,\longrightarrow\,R(a)[/tex] Universal instantiation of (4)

6) [tex]R(a)[/tex] Modus Ponens of (5)

Here I am stuck, any suggestions?
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