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

Here is one way:

$\begin{align} P= \{ &102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, \\ &136, 138, 140, 142, 144, 146, 148, 150, 152, 154, 156, 158, 160, 162, 164, 166, 168, \\ &170, 172, 174, 176, 178, 180, 182, 184, 186, 188, 190, 192, 194, 196, 198, 200, 202, \\ &204, 206, 208, 210, 212, 214, 216, 218, 220, 222, 224, 226, 228, 230, 232, 234, 236, \\ &238, 240, 242, 244, 246, 248, 250, 252, 254, 256, 258, 260, 262, 264, 266, 268, 270, \\ &272, 274, 276, 278, 280, 282, 284, 286, 288, 290, 292, 294, 296, 298, 300, 302, 304, \\ &306, 308, 310, 312, 314, 316, 318, 320, 322, 324, 326, 328, 330, 332, 334, 336, 338, \\ &340, 342, 344, 346, 348, 350, 352, 354, 356, 358, 360, 362, 364, 366, 368, 370, 372, \\ &374, 376, 378, 380, 382, 384, 386, 388, 390, 392, 394, 396, 398, 400, 402, 404, 406, \\ &408, 410, 412, 414, 416, 418, 420, 422, 424, 426, 428, 430, 432, 434, 436, 438, 440, \\ &442, 444, 446, 448, 450, 452, 454, 456, 458, 460, 462, 464, 466, 468, 470, 472, 474, \\ &476, 478, 480, 482, 484, 486, 488, 490, 492, 494, 496, 498 \} \end{align}$.

Another slightly shorter way:

$P= \{ n \in \mathbb{N} | 100 < n < 500 \text{ and } \sin ( n {\pi \over 2} ) = 0 \}$.

Inspired by Charlotte's answer:

$P= (2 \mathbb{N}+\{100\}) \setminus (2 \mathbb{N}+\{498\})$.

Et iterum (Haskell's take on Ross' answer):

[2*x | x <- [51..249] ]

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.

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).