Question

Suppose that the domain of the propositional function P(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) ∃xP(x)
b) ∀xP(x)
c) ¬∃xP(x)
d) ¬∀xP(x)
e) ∀x((x=3) → P(x))∨∃x¬P(x)

Answer 1

a) $∃xP(x)=P(1)\lor P(2)\lor P(3)\lor P(4)\lor P(5).$

b) $∀xP(x)=P(1)\land P(2)\land P(3)\land P(4)\land P(5).$

c) $¬∃xP(x)=¬(P(1)\lor P(2)\lor P(3)\lor P(4)\lor P(5))$

d) $¬∀xP(x)=¬(P(1)\land P(2)\land P(3)\land P(4)\land P(5)).$

e) $∀x((x=3) → P(x))∨∃x¬P(x)=∀x(¬(x=3) \lor P(x))∨∃x¬P(x)=$ $(¬(1=3) \lor P(1))\land (¬(2=3) \lor P(2))\land (¬(3=3) \lor P(3))\land (¬(4=3) \lor P(4))\land (¬(5=3) \lor P(5))\lor ¬P(1)\lor ¬P(2)\lor ¬P(3)\lor ¬P(4)\lor ¬P(5)$