Question

Suppose the domain of the propositional function P(x) consists of the integers −2,

−1, 0, 1, 2. Write out each of these propositions using disjunctions, conjunctions, and

negations.

a) ∃xP(x)

b) ∀xP(x)

c) ∃x¬P(x)

d) ∀x((x ≠ 1) → ¬P(x))

Answer 1

One of the ways is presented below

a). $\exists x P(x)$ can be rewritten as: $P(-2)\lor P(-1)\lor P(0)\lor P(1)\lor P(2)$

b). $\forall P(x)$ can be rewritten in the following form: $P(-2)\land P(-1)\land P(0)\land P(1)\land P(2)$

c). $\exists x\lnot P(x)$ can be rewritten as: $\lnot(P(-2)\land P(-1)\land P(0)\land P(1)\land P(2))$

d). $\forall x((x\neq1)\rightarrow\lnot P(x))$ can be rewritten as: $\lnot((\lnot(-2\neq-1)\lor P(-2))\land(\lnot(-1\neq-1)\lor P(-1))\land(\lnot(0\neq-1)\lor P(0))\land(\lnot(1\neq-1)\lor P(1))\land(\lnot(2\neq-1)\lor P(2)))$