Question

# Use De Morgan’s Laws to distribute negations inward (that is, all negations in the resulting equivalent...

Use De Morgan’s Laws to distribute negations inward (that is, all negations in the resulting equivalent expression should appear before a predicate): ¬∃x(¬∀y(Q(x,y) v ¬R(x,y)) → ∀yP(x,y)). Label each step with the name of the rule used (such as DN: Double Negation, see formula sheet) and which statement numbers the rule uses. The first step is: 1. ¬∃x(¬∀y(Q(x,y) v ¬R(x,y)) → ∀yP(x,y))

1. ¬∃x(¬∀y(Q(x,y) v ¬R(x,y)) → ∀yP(x,y))
2. ∀x¬(¬∀y(Q(x,y) v ¬R(x,y)) → ∀yP(x,y)) [DeMorgan's]
3. ∀x¬(¬¬∀y(Q(x,y) v ¬R(x,y)) v ∀yP(x,y)) [Implication]
4. ∀x¬(∀y(Q(x,y) v ¬R(x,y)) v ∀yP(x,y)) [Double negation]
5. ∀x(¬∀y(Q(x,y) v ¬R(x,y)) ∧ ¬∀yP(x,y)) [DeMorgan's]
6. ∀x(∃y¬(Q(x,y) v ¬R(x,y)) ∧ ¬∀yP(x,y)) [DeMorgan's]
7. ∀x(∃y(¬Q(x,y) ∧ ¬¬R(x,y)) ∧ ¬∀yP(x,y)) [DeMorgan's]
8. ∀x(∃y(¬Q(x,y) ∧ R(x,y)) ∧ ¬∀yP(x,y)) [Double negation]

Answer: ∀x(∃y(¬Q(x,y) ∧ R(x,y)) ∧ ¬∀yP(x,y))

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