**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)**

The **Answer to the Question**

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**Here's the Solution to this Question**

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