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?

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).