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)

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

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Question ID: mtid-5-stid-8-sqid-3509-qpid-2208