Answer to Question #147067 in Discrete Mathematics for taha

Question #147067
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)
1
Expert's answer
2020-11-27T18:22:32-0500

a) "\u2203xP(x)=P(1)\\lor P(2)\\lor P(3)\\lor P(4)\\lor P(5)."


b) "\u2200xP(x)=P(1)\\land P(2)\\land P(3)\\land P(4)\\land P(5)."


c) "\u00ac\u2203xP(x)=\u00ac(P(1)\\lor P(2)\\lor P(3)\\lor P(4)\\lor P(5))"


d) "\u00ac\u2200xP(x)=\u00ac(P(1)\\land P(2)\\land P(3)\\land P(4)\\land P(5))."


e) "\u2200x((x=3) \u2192 P(x))\u2228\u2203x\u00acP(x)=\u2200x(\u00ac(x=3) \\lor P(x))\u2228\u2203x\u00acP(x)=" "(\u00ac(1=3) \\lor P(1))\\land (\u00ac(2=3) \\lor P(2))\\land (\u00ac(3=3) \\lor P(3))\\land (\u00ac(4=3) \\lor P(4))\\land (\u00ac(5=3) \\lor P(5))\\lor \u00acP(1)\\lor \u00acP(2)\\lor \u00acP(3)\\lor \u00acP(4)\\lor \u00acP(5)"



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