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Let $P(x) : x^2 \leq 4$. The domain for $x$ is all positive integers. Determine truth values of the following propositions

(a) $P(1)$

(b) $\exists x \neg P(x)$

for (a)

I got $P(1)$ is the proposition $(1)^2 \leq 4$. Since $1<=4$ is true, it has a truth value T.

for (b)

I am unsure how to go about this. I started with creating an equivalency but got stuck.

$\exists x \neg P(x) \iff \neg[\forall x P(x)]$

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Your response to (a) is correct. For (b), it is just asking if there exists some positive integer such that $P(x)$ is false. This is a true statement. Indeed, consider $x=3$.

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