Expressing a number that doesn't exist

Question

How can one express something like $x \in \pi$ where $\pi$ is a set of prime numbers and $d$ is some divisor such that $\pi = \lbrace n:d|n\rbrace = \lbrace {1, p}\rbrace$? Or should I do something like this $\pi = \lbrace p:d|p=\lbrace1, p\rbrace\rbrace$? Thanks for the help in advance.

Answer 1

There are lots of ways to express the fact that $\pi$ is the set of prime numbers. One is just $$ \pi = \{p\in\mathbb{N} : p\text{ is prime}\}. $$ More explicitly, you could write $$ \pi = \{p\in\mathbb{N} : \forall_{d\in\mathbb{N}}(d\mid p\rightarrow d = 1\vee d=p)\} $$ or $$ \pi = \{p\in\mathbb{N} : \neg\exists_{d\in\mathbb{N}-\{1,p\}} (d\mid p) \}. $$

Answer 2

I would just express it as $\{ p \in \mathbb{N} | p\text{ is prime} \}$. Every positive integer larger than $1$ has at least two divisors: $1$ and itself, so your alternatives are false. If you really want to express it in terms of divisors, you could write: $\{ p \in \mathbb{N} | p \text{ has exactly two distinct divisors} \}$.

Answer 3

So apparently one of the correct ways is this: $$\lbrace p\in \mathbb N : p \in\lbrace n:n|p\rbrace = \lbrace 1,p\rbrace\rbrace$$.