Answer to Question #147067 in Discrete Mathematics for taha

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