Using set notation, define the set of even natural numbers between 100 and 500. 
Using set notation, define the set of even natural numbers between 100 and 500.

This is what I have so far:
$P$ is even numbers so the set of natural numbers between 100 and 500 would be
$$P = \{x:x \in\mathbb  N, 100 < x < 500\}$$
Would this be correct?
 A: The most succinct way I know is
$$\{x|100\leq x \leq 500, x\in2\mathbb{Z}\}$$
Inspired by copper.hat and NicolajK, inter alia, here are some further valid answers:
$$\left\{2x+100\Bigg|\prod_{i=0}^{200}(x-i)=0\right\}$$
$$\{x|x\text{ is an even integer between 100 and 500}\}$$
$$\left\{x\Bigg|x=\frac{p^{(2n)}(0)}{p^{(2n-1)}(0)}, n\in\mathbb{N}\right\},\quad p(x)=\sum_{i=100}^{500}x^i$$
$$\{x|x=\log_2[G:H],\,H\subseteq G,\,|H|\leq 2^{100},\,|G|=2^{500}\},
\quad G\text{ is a group}$$
$$\begin{aligned}&\{((q\text{ incr})0)|\exists p\in\mathbb{H},
\\&q=
\\&\lambda pfx.((\lambda MNfx.N(Mf)x)(\lambda MNfx.(NM)fx)f(f(f(f(fx))))f(f(x)))f((pf)pfx)
\\&\text{and}
\\&\lambda M.\lambda N.(\lambda n.n(\lambda x.(\lambda ab.\lambda b))
(\lambda ab.a))(\lambda mn.n(\lambda nfx.n(\lambda gh.h(fg))(\lambda u.x)(\lambda u.u)\\& m) M N)q(\lambda MNfx.(NM)fx)f(f(f(f(fx))))f(f(f(x)))f((pf)pfx)\}\end{aligned}$$
where $\mathbb{H}=\{\lambda fx.x,\lambda fx.fx,\lambda fx.f(fx),\lambda fx.f(f(fx)),\dots\}$.
A: For something very close to your proposal, you could say $$P=\{2x:x \in \Bbb N, 50 \lt x \lt 250\}$$  The $2x$ is one way to get rid of the odd numbers.
A: Computer scientists know that any binary number that is greater than zero is even, if the least significant bit is zero
$$ \{x:x \in \Bbb N, 100 \lt x \lt 500, x=(b_8\dots b_10)_2\} $$
Cryptographers might do it like this
$$ \{x:x \in \Bbb N, 100 \lt x \lt 500, x\equiv 0\ (\mathrm{mod}\ 2)\} $$
A: Some more versions:


*

*$\{x\in 2\Bbb N|100<x<500\}$ or $\{x\in 2\Bbb Z|100<x<500\}$

*$2\Bbb N \cap (100;500)$ or $2\Bbb Z\cap (100;500)$
A: Using remainder classes, you can express it as
$$
P = \lbrace 100 + [n]_{400} \mid n \in \mathbb{N} \rbrace
$$
Which is equal to
$$
P = 100+\mathbb{Z}_{400}
$$
A: If we really want short, we could do this: 
$$2\mathbb Z\cap[100,500]$$
If you need the set brackets:
$$\{x:x\in 2\mathbb Z\cap[100,500]\}$$
where $[a,b]=\{x:x\in \mathbb R \land a \le x \le b\}$, the normal sense of $[a,b]$
A: Let $\mathbb N=\{1,2,3,...\}$. You want a set which includes the even members of $\mathbb N$ that lie between 100 and 500. Well, a member $n$ of $\mathbb N$ is even precisely when $n=2k$ for some $k\in\mathbb N$.
So $\{n\in\mathbb N:(\exists k\in \mathbb N)(n=2k)\text{ and } 100<n<500\}$ works.
(use the weak inequality $\leq$ if you want to include 100 and 500 in the set).
A: I would tend to use one of;
$P = \{ n \in \mathbb{N}: 100 < n < 500, 2\mathop{|}n \}$
or
$P = \{n \in \mathbb{N}: \text{$n$ even}, 100 < n < 500 \}$
The latter I suppose is slightly less formal, but it would be perfectly normal to see it used in a lecture or talk, say, and it is probably the clearest possible when speaking or reading.
Note $m \mathop{|} n$ means that $m$ divides $n$, i.e. $n$ is an integer multiple of $m$.
A: A few more ways to express it:
$P=\{x\in\mathbb{N}:100<x<500\land\lfloor\frac{x}{2}\rfloor=\frac{x}{2}\}$
$P=\{x\in\mathbb{N}:100<x<500\land x\equiv0\mod{2}\}$
$P=\{x\in\mathbb{N}:100<x<500\land\frac{x}{2}\in\mathbb{N}\}$
A: $\{|n| \mid n\in E, 100<|n|<500\}$
where
$E\equiv\{\emptyset\}\cup\{x\mid\exists (y\in E).\,x=\left(y\cup\{y\}\cup\{y\cup\{y\}\}\right)\}$
A: Here is another way that has not yet been suggested here:
$$P=\{n\in\mathbb{N}|[100<n<500]\wedge[(-1)^n=1]\}$$
A: I think this is a common way of listing a set of even numbers:
$P=\{2x + 100:x\in\mathbb{N}, 0\leq x\leq 200\}$
This just means:
Let P be the set of numbers yielded from "2x + 100" where x is a natural number and x is between 0 and 200 (inclusive). 
If you start evaluating 2x + 100 with the numbers from 0 to 200, you get:


*

*2(0) + 100 = 100  

*2(1) + 100 = 102 

*... 

*2(200) + 100 = 500

A: $$ \left\{ x \mid x \in \mathbb{N},\ 2 \mid x,\ 100 \leq x \leq 500 \right\} $$
Since you haven't specified, I'm assuming between 100 and 500 is inclusive.
$2 \mid x$ means $2$ is a divisor of $x$, or alternatively, $x$ is evenly divisible by $2$.
A: This is a more longwinded approach to help clarify what's going on. But be explicit in how you define 'between'. In this sense, between could be interpreted as $100 \leq x \leq 500$ or $100 < x < 500$. As long as you are explicit about your assumption, you're fine.
The sentence "Define the set of even natural numbers between 100 and 500" can be broken into several components:

*

*P = {even natural numbers between 100 and 500}

*P = {elements of x in natural numbers such that (x is even) AND (x is greater than 100) AND (x is less than 500) }

*P = $ \{ x \in \mathbb{N} : (\exists k \in \mathbb{N} \quad x = 2k) AND (x > 100) AND (x < 500) \} $

*P = $ \{ x \in \mathbb{N} : (\exists k \in \mathbb{N} \quad x = 2k) \land [(x > 100) \land (x < 500)] \} $
This way makes the inequality $100 < x < 500$ explicit using the logical and operator.
