How to notate the universal quantifier when applied to an equation?
So I have an equation with infinite regular solutions. Let's say this equation is $$\sin^{-1}(0)=\pi n$$ where $n$ is any integer. How do I express this formally using the universal quantifier?
Do I say
- $\forall n\in\mathbb Z: \sin^{-1}(0)=\pi n$
- $\sin^{-1}(0)=\forall n\in\mathbb Z:\pi n$
- $\sin^{-1}(0)=\pi n, \forall n\in\mathbb Z$
Or is there a more aesthetic way to express this?
In other words, by convention, where am I supposed to put the universal quantifier when an equality sign is involved?
Using natural language (like you did in the first paragraph of your question) is a very common, and in many cases, the preferred way of writing a quantification.
But there are occasions where using symbols is preferable. Formal definitions of formulas usually dictate that the quantifier must come first, i.e. $$ \forall x \in X: f(x) = g(x). $$ If you work with formulas in logic or model theory, this is the only way.
Outside of this context, it can be more pleasant to put the formula up-front, and most people would find $$ f(x) = g(x) \qquad \forall x \in X $$ acceptable as well. It is generally a bad idea to mix quantifiers before and after the formula, though, because there is no convention in which order they apply (and $\forall$ and $\exists$ cannot be interchanged in general).
Your second option really makes no sense at all because it looks like $\sin^{-1}(0)$ is somehow equal to the “statement” $\forall n \in \mathbb Z : \pi n$ (which isn’t even a statement because $\pi n$ isn’t).
$$\sin(x)=0\iff\exists\ n\in\mathbb Z:x=n\pi.$$