Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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? \} $

Thanks in advance, Regards, Misery

share|improve this question
2  
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
1  
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
1  
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
1  
$\{n \in \mathbb N: n \text{ is even}\}$ –  sdcvvc May 12 '12 at 8:03

1 Answer 1

up vote 4 down vote accepted

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.