How does one write "For every multiple of $n$, $f(n)=0$" in set builder notation?

Question

I'm actually writing about the characteristics of a periodic function and I've stumbled a (not really) problem.

I wanted to tell the readers that,

For all $x$, $\sin(x)=0$, such that $x$ is a multiple of $2\pi$,

but writing this in a plain form is quite awkward because normally, mathematicians uses symbols (and maybe it's even wrong!).

Is there a compact notation of this in mathematics using the set builder notation?

Answer 1

Other answers have addressed how one might use notation and have noted that $\sin(x) = 0$ if and only if $x$ is an integer multiple of $\pi$ (i.e. not just an integer multiple of $2\pi$). This answer seeks to address (1) the awkward phrasing of the statement for which the asker seeks notation, and (2) a misconception about the use of notation. I am specifically not answering the question in the title, which asks for set builder notation. Consider this a frame challenge.

Beginning with the use of notation, the asker asserts that "mathematicians uses symbols". Let me start by disabusing the asker of that misconception: the goal of mathematical writing is to convey an idea in the clearest way possible. Sometimes, notation does this best. Other times, plain language does this best. Most often, it is a combination of notation and plain language which best conveys ideas. Do not assume that your writing is better because it uses more notation, and don't throw in notation for notation's sake. Again, the goal is to communicate.

Regarding the specification of a set, the asker writes

for all $x$, $\sin(x)=0$, such that $x$ is a multiple of $2\pi$.

This looks like gibberish to me. It first says that $\sin(x) = 0$ for all $x$, then restricts the values of $x$ which are considered. If I'm reading this, I am going to get as far as "for all $x$, $\sin(x) = 0$" and immediately become confused. Either tell us which $x$ are being considered first (i.e. $x$ such that $x$ is an integer multiple of $2\pi$), or give the identity first and then explain which $x$ satisfy the identity. For my taste, I might write

for all $x$ such that $x$ is an integer multiple of $2\pi$, we have $\sin(x) = 0$.

Alternatively (and I think this reads a little better),

for all $x = 2k\pi$ where $k\in\mathbb{Z}$, $\sin(x) = 0$,

or even

if $x = 2k\pi$ for some integer $k$ then $\sin(x) = 0$.

Personally, I think that the last version reads best (though it might not quite be what is desired in the context of the question): it mixes notation and plain English in a manner which flows reasonably well and creates no ambiguities.

Answer 2

You could write $\sin(2\pi k)=0$, for every integer $k$.

Answer 3

It's not always the best to use symbols for mathematics. The best is often a good mixture, like writing "$\sin(x)$ has zeros at all multiples of $\pi$".

Answer 4

Precisely speaking, this would be written as $$\sin( x) =0\ \text{if } x\in \{n\pi \ |\ n\in \mathbb{Z}\}$$ However it would also be perfectly clear to say "$\sin(x)$ has roots at integer multiples of $\pi$.".