Answer to Question #327818 in Discrete Mathematics for Gotdon

Question #327818

What are the truth values of these statements?


[3 marks]


a) ∃!xP(x)→∃xP(x)


b)


∀x P(x) → ∃!xP(x)


c)


∃!x¬P(x)→¬∀xP(x)

1
Expert's answer
2022-04-17T10:01:46-0400

a). The statement !xP(x)\exists!xP(x) means that there is a unique xx that satisfies the statement P(x)P(x). From the latter it follows that there is xx that satisfies P(x)P(x). It means that the statement !xP(x)xP(x)\exists!xP(x)\rightarrow\exists xP(x) holds. Thus, !xP(x)xP(x)\exists!xP(x)\rightarrow\exists xP(x) is true.

b). The statement xP(x)\forall x P(x) means that P(x)P(x) holds for all xx, whereas !x\exists!x means the existence and uniqueness of xx . Thus, the statement xP(x)!xP(x)\forall x P(x)\rightarrow\exists!x P(x) is false.

c). The statement !x¬P(x)\exists!x\lnot P(x) means that there is a unique xx that does not satisfy P(x)P(x) (satisfies the negation of P(x)P(x) ). Therefore, the statement P(x)P(x) does not hold for all xx. In other words, ¬xP(x)\lnot\forall x P(x). Thus, the statement !x¬P(x)¬xP(x)\exists!x\lnot P(x)\rightarrow\lnot\forall x P(x) is true.


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