Correct set notation for "all integers which are not multiples of 7"? What is correct set notation for "all integers which are not multiples of $7$"? My best guess is:
$$ \{ x : (\forall k \in \mathbb{Z})(\neg(7k = x)) \}$$
Or
$$ \{ x : \neg(\exists k \in \mathbb{Z})(7k = x) \}$$
However this seems unlike other examples I have seen.
Is there are proper way to denote this set in set notation?

I think the vertical line should be used in sets. This is how it's usually done:
$$\{x \in \mathbb{Z} \mid \forall z \in \mathbb{Z}: x \neq 7z \}$$
First part naming some elements, then after the pipe comes the conditions for those elements. I actually have never seen the colon used instead of the pipe before in any lecture material or mathematical book. I think their usage is different.
$$\{x \in \mathbb{Z} \mid 7 \nmid x \}$$
Is another shorter form, but as others said, it required knowledge of the $\nmid$ symbol.
