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?
$\endgroup$ 99 Answers
$\begingroup$I'd generally go with $\{x\in\mathbb{Z}: 7 \nmid x\}$, though this does presuppose familiarity with the $\mid \, \nmid$ notation.
$\endgroup$ 1 $\begingroup$For starters, you should be using curly braces $\{\}$ for sets, and what you have is fine, but generally the clean abbreviation would be: $$ \mathbb{Z}-7\mathbb{Z}$$
This is because $-$ denotes set difference and $7\mathbb{Z}$ denotes the set of integer multiples of $7$.
$\endgroup$ 9 $\begingroup$My preferred notation is the "not divides" symbol $a \nmid b$ what means "a not divides b". But you can achieve the same meaning by different ways, by example:
$$\{n\in\Bbb Z: (\nexists k\in\Bbb Z :n/k=7)\}$$
or
$$\{n\in\Bbb Z:n\not\equiv 0\pmod 7\}$$
But the common notation is
$$\{n\in\Bbb Z:7\nmid n\}$$
$\endgroup$ 2 $\begingroup$Of course, you could always use $$\{ x \in \mathbb{Z} : \text{$x$ is not a multiple of $7$} \}$$
If you think the notation will get in the way of what you're trying to say, then discard the notation!
$\endgroup$ $\begingroup$The most accurate translation of the English condition would be $$ \left\{n\in\mathbb{Z}:\frac n7\not\in\mathbb{Z}\right\} $$
$\endgroup$ 1 $\begingroup$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.
$\endgroup$ 3 $\begingroup$Another possibility (more wordy but to me clearer):
$$\{x \in ℤ:x \neq 7k \quad \forall k∈ℤ\}$$
$\endgroup$ 5 $\begingroup$The only issue with what you wrote is that you didn't specify what set the $x$ should come from. For example, $x$ could be $1.5$, since $1.5$ isn't an integer multiple of 7. More correct would be
$$ \{ x\in \mathbb{Z} : (\forall k \in \mathbb{Z})(\neg(7k = x)) \}$$
Of course, in terms of simplicity, as others have noted, there are more compact notations for divisibility that you can use.
$\endgroup$ $\begingroup$What about: {7j+k (∀j∈Z) (∀k∈{1,2,3,4,5,6})} ?
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