I was wondering if this expression, $f$, could be bounded by some constant for any set $S,T \subseteq \Omega$, or at least argue about the sign of this expression.(I'm looking for a bound tighter than simply assuming every sine and cosine is equal to 1)
$S,T \subseteq \Omega$ where $\Omega$ is a set of $N$ elements. $g_k$ is a real valued function of $k$ where $k$ is an element of the set.
$f = \sum_{∀k∈S} cos(g_k) $ $\sum_{∀k∈T} cos(g_k) - \sum_{∀k∈S} cos(g_k) $ $\sum_{∀k∈S∩T} cos(g_k)- \sum_{∀k∈T} cos(g_k) $ $\sum_{∀k∈S∩T} cos(g_k)+\sum_{∀k∈S} sin(g_k) $ $\sum_{∀k∈T} sin(g_k)-\sum_{∀k∈S} sin(g_k) $ $\sum_{∀k∈S∩T} sin(g_k)-\sum_{∀k∈T} sin(g_k) $ $\sum_{∀k∈S∩T} sin(g_k)$
Yes it is bounded.
$|\cos(x)|,|\sin(x)|\leq 1$ By the triangle inequality we can replace each sin and cos by $1$ we have
$$|f| \leq |S|+|T|+|S||S\cap T|+|T|+|S\cap T|+|S||T|+|S|+|S\cap T|+|T||S\cap T|$$
so $$|f|\leq 2|S|+2|T|+2|S\cap T| + (|T|+|S|)(|S\cap T|)$$
You can't tell the sign for sure, because it depends on $g_k$.