Prepositional Logic

Question

Suppose that the domain of the propositional function P(x) consists of the integers − 2, −1, 0, 1, and 2.

Write out propositions ∀x¬P(x) and ¬∃xP(x) using disjunctions, conjunctions, and negations.

Answer 1

 

$ \forall x\neg P(x) = \neg P(−2)\wedge \neg P(−1)\wedge \neg P(0)\wedge \neg P(1)\wedge \neg P(2)$

Applying De Morgan’s law, we get $ \forall x\neg P(x)=\neg\exists x P(x)$

$ \neg\exists x P(x) = \neg(P(-2) \vee P(-1) \vee P(0) \vee P(1) \vee P(2))$