Chapter 1.4, Problem 19E

Suppose that the domain of the propositionalP(x) consists of the integers 1, 2, 3, 4, and 5. Express these statements without using quantifiers, instead using only negations, disjunctions, and conjunctions.

a) $\exists xP\left(x\right)$

b) $\forall xP\left(x\right)$

c) $\neg \exists xP\left(x\right)$

d) $\neg \forall xP\left(x\right)$

e) $\forall x\left(\left(x\ne 3\right)\to P\left(x\right)\right)\vee \exists x\neg P\left(x\right)$