Here is what I am attempting to say is:
"For every b in B, there is a set of r's such that for every v in V, there exists exactly one r"
$$\forall b \in B \{r \ | \ \forall v \in V, \exists ! r \}$$
In a similar vein, is it correct to say:
$$ X = \{y = \sum j(k) \ | \ \forall c \in C, \exists ! y\}$$
Here, I am trying to say:
"There is some set X that is made up of y's (which are calculated via the summation of j(k) ) such that for every c in C, there exists only one y."
I tried the Wikipedia page for set builder notation, but all that did was confuse me even more.
Another way to frame it is:
I have a set of things that I value (V). For example, lets say SEX, DRUGS, and ROCK-AND-ROLL (or v1, v2, v3 if you prefer). I also have, at any given moment, a set (B) of actions (b1, b2, b3) that I can undertake. Every possible action in the set of all possible actions will yield exactly one return (r) for everything that I value. That is, every b from the set B will generate exactly one r for every v in V.
I'll take a stab at this given the extra information in the comment (which should be edited into the question).
I'll assume the "return" $r$ is a member of a set $R$ (which might be a nonnegative real number, but doesn't have to be). Then I think what you are describing is a function that assigns a member of $R$ to each pair $(v,b)$ where $v \in V$ and $b \in B$.
I'd make the names of the sets more descriptive - say WISHES and ACTIONS for $V$ and $B$, VALUES for $R$ and a function $$ \text{return}: \text{WISHES} \times \text{ACTIONS} \to \text{VALUES} $$