# How to write the set of all natural numbers, that can be divided by two?

I would like to write as a mathematical expression such sentence: x is the set of all natural numbers that can be divided by 2 without the remainder.

$\{x \in \mathbb N: What \ here? \}$

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The simplest (and to my mind the nicest) way is $\{2n:n\in\Bbb N\}$. An alternative is $\{n\in\Bbb N:2\mid n\}$, since $a\mid b$ means '$b$ is a multiple of $a$'. –  Brian M. Scott May 12 '12 at 7:42
Note that if $x$ is "the set of all natural numbers that can be divided by $2$", then $x$ is the set, i.e. $$x = \{ n \in \mathbb N \, | \, 2 \text{ divides } n \}$$ –  Patrick Da Silva May 12 '12 at 7:43
In other words, $x$ is the set, he's not an element of the set like you just wrote in your question. If you want the set of all elements $x \in \mathbb N$ such that blablabla, then you expect the set to look like $\{ x \in \mathbb N \, | \, \text{ blablabla here. } \}$. –  Patrick Da Silva May 12 '12 at 7:44
I found that notation in the book saying that: {x: P(x)} means the set of those all x that have the property P, and I got confused trying to write the set as given in the question –  Misery May 12 '12 at 7:47
$\{n \in \mathbb N: n \text{ is even}\}$ –  sdcvvc May 12 '12 at 8:03
A very typical shorthand when writing the set of natural numbers divisible by $a$ is to write $a\mathbb N = \{an \in \mathbb N\;|\; n \in \mathbb N\}$. Clearly every natural number of the form $an$ where $n\in \mathbb N$ is divisible by $a$ and every such number arises in this way. This follows the usual notation for cosets.