I don't have a real background in math but I should still be able to define stuff in my MSc thesis, although the thesis does not involve a lot of math.
I want to define an object $o$'s spatiotemporal routine $R_{o}$ as a set of frequently visited places $p_{i}(t)$ by this object. The visits to each place are dependent on time, i.e. (mostly) visited during certain times (of the day, of the week, does not really matter). I cannot and don't want to define "frequent" yet, thus I somehow want to say that the frequency of visits should be above a threshold $\alpha$. I came up with this definition:
$$R_{o} = \left\{p_{1}(t),p_{2}(t),p_{i}(t),...,p_{n}(t)|\forall p_{i}: freq(p_{i}) \geq \alpha\right\}$$
My questions: 1. Is this a correct use of the $\forall$ quantifier? It should mean that the frequency criterion should apply to ALL places which are contained in the set. 2. Is there another way to define frequency than to use the ugly $freq()$?
Maybe somebody can come up with a better, more elegant, or more "professional" definition, I've never done that before but it shouldn't be too difficult, right?
It appears you want to say $$R_o=\{\,p(t)\mid \operatorname{freq}(p(t))\ge \alpha\,\}$$