Chapter 1.4, Problem 20E

Suppose that the domain of the propositionalP(x) consists of the integers -5, -3, -1, 1, 3, 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)

   $\forall x\left(\left(x\ne 1\right)\to P\left(x\right)\right)$

d) $\exists x\left(\left(x\ge 0\right)\wedge P\left(x\right)\right)$

e) $\exists x\left(\neg P\left(x\right)\right)\wedge \forall x\left(\left(x<0\right)\to P\left(x\right)\right)$