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I am working through "How to prove it: a structured approach".

4.Write definitions using elementhood tests for the following sets:

a) $\{ 1,4,9,16,25,36,49,\ldots \}$

For a. my first intuition was $\{x ∈ \Bbb{N}\ |\ x^2\}$ but then I realized it would not constitute a "test". So my question is what is the best way to express such a thing mathematically? My best attempt so far is $\{x | \sqrt{x} ∈ \text{positive integers}\}.$

Second point of confusion is on some website I found this answer which seems to introduce free variable $y$ making it true or false depending which $y$ is chosen. Basically if this following answer is correct, then I am REALLY confused. $\{x ∈ \Bbb{R}\ |\ x = y^2 \text{ for some } y ∈ \Bbb N\}.$

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    $y$ is not a free variable. Yes, it is free in the equation "$x = y^2$", but in the larger context it is bound by the "for some $y \in \mathbb N$" that follows. So the statement "$x=y^2$ for some $y\in\mathbb N$" is not true or false *depending on which $y$ is chosen*, it is true if it is at all possible to choose a $y$ such that $x=y^2$.2012-07-29

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