I have two functions of time: $f(t)$ and $g(t)$.
These functions oscillate over time at different rates.
I am hoping to build a set, $S$, that contains all of the times at which $f(t) = X$, as well as those times at which $g(t) = X$. Below is my attempt at building this set.
$S=\{t | \forall i \in \{f, g\}: i(t)=X\}$.
Is this correct? Is there an easier way to build this set?
On
It's a little unconventional to quantify over a set of functions that way. Letting variables take on functional values like that smells a little like programming to me. But I don't see anything inherently wrong about it.
However, you do want $\exists$ instead of $\forall$. You don't want the $t$ such that $f(t)=X$ and also $g(t)=X$. You want the either-or(-or-both) variation, the way I read your question: the $t$ such that at least one among $f$ and $g$ evaluates to $X$.
If you had an indexed list of functions $\{f_1,f_2,\ldots\}$ (either infinite, or finite, possibly with only two functions), then $$ \{t\mid \exists i(f_i(t)=X)\} $$ would be entirely standard. Quantifying over indices in this manner is much more common than quantifying over the actual functions. I don't know exactly why.
In your case, the most conventional would probably be something like $$ \{t\mid f(t)=X\text{ or }g(t)=X\} $$
This describes the set of all times $t$, for which $f(t) = g(t) = X$, but reading your description I think you want the set of all times $t$ for which either $f(t) =X$ or $g(t) = X$. In your notation, this is $$ S = \{t : \exists i \in \{f,g\}. i(t) = X\}. $$ I also think that this is quite an unusual ansatz to write this set, I'd write something like $$ S = \{t: f(t) = X \vee g(t) = X \} $$ or even, using the preimage operation $$ S = f^{-1}[\{X\}] \cup g^{-1}[\{X\}]. $$