# Right notation for an interval in set builder notation?

So I'm trying to figure out how to correctly display a natural number that lies in an interval and when searching for it, I get a variety of ways and so I'm unsure about the correct one

So say we have something like all even natural numbers between 0 and 10.

The way I did it: {$${2x | x \in N∧ 0 \leq x \leq 5}$$}

But then, others write it as {$${2x | x \in N}, 0 \leq x \leq 5$$} or some even use the "therefore |" sign in different ways or skip the "therefore" at the beginning all together and add it after the set definition so its all a bit much.

I'm really unsure about how to define sets properly, would really appreciate some (beginner friendly) help. Thanks in advance!

There are a lot of potential ways to represent this set, all of which are valid. In the two examples you have listed, the only difference is the way that the two properties are separated. You could also just use a word to separate them like $$\{2x|x\in N and\, 0\leq x\leq5\}$$. When I define sets, I like to use a colon : instead of the vertical line | so it looks like $$\{2x:x\in N and\, 0\leq x\leq5\}$$. I like the way this looks better, especially when written by hand where the | can look like a 1. You could also write it as $$\{x|x$$ is even and $$0\leq x\leq10\}$$.