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)
a). The statement means that there is a unique that satisfies the statement . From the latter it follows that there is that satisfies . It means that the statement holds. Thus, is true.
b). The statement means that holds for all , whereas means the existence and uniqueness of . Thus, the statement is false.
c). The statement means that there is a unique that does not satisfy (satisfies the negation of ). Therefore, the statement does not hold for all . In other words, . Thus, the statement is true.
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