Correct notation to restrict parameter in equation

Question

I'm trying to express the following equation using correct notation:

$\sin{\left(\frac{n\pi}{2}\right)},n\,\text{even} = 0$

I've already specified $n$ is a natural number, so presumably I don't need to respecify it? Would the following be better?

$\left\{\sin{\left(\frac{n\pi}{2}\right)}:n\,\text{even}\right\} = 0$

Is there a more "mathy" way to express that $n$ is even? What if I hadn't yet specified that $n$ belonged to the natural numbers, would I use something like?

$\left\{\sin{\left(\frac{n\pi}{2}\right)}:n\in\mathbb{N}\,,\text{even}\right\} = 0$

Are all of these essentially acceptable? None of them?

Answer 1

I guess you'd like to say that $$\sin\left(\frac{n\pi}{2}\right) = 0$$ for all even $n \in \mathbb{N}$. Am i correct?

You can indeed express the set of all these values via

$$\left\{\sin\left(\frac{n\pi}{2}\right) \mid n \in \mathbb{N},\ n\ \text{even}\right\} = \{0\}.$$

Note that your expression

$$\left\{\sin\left(\frac{n\pi}{2}\right) \mid n\ \text{even}\right\} = 0$$

is incomplete, since you don't mention that $n \in \mathbb{N}$ and on the left hand side you have a set, whilst on the right hand side we have a number.

Note: This is something you might sometimes encounter in abstract algebra where the trivial group $\{e\}$ is written as $0$ or $1$. But in a scenario like this, i would not recommend it.