Question

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)

Answer 1

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

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

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