Tell me if I'm saying this right.
A Cartesian Product is a function $f:X \times Y \to Z$ , where some unknown structural operation on the sets $X$ and $Y$ produces a set $Z$ as its codomain, and $Z$ is a set of ordered pairs $(x,y)$ where $x \in X$ and $y \in Y$ for all possible values of $x$ and $y$. And codomain, while kind of arbitrarily defined, is generally the set of all possible values output by $f$, the set of ordered pairs $(x,y)$.
Good, so far?
Now, tell me if I'm describing binary operations right.
Now a binary operation, it seems, is a cartesian product $f: Y \times Y \to Y$, where $f$, an unknown structural operation, operates on the sets $Y$ and $Y$ (which are for all purposes identical) and produces a set $Y$ (identical to the first two) as its codomain. Codomain has the same definition has earlier.
Okay, here's where I get really confused regarding binary operations. From the first definition, it seems that a Cartesian Product is always a set of ordered pairs, but a binary operation is not producing ordered pairs of elements, but single elements that are also members of the set from which the two elements being operated on are members.
Could anyone help me make sense of, help me bridge this—how do I say?—cognitive dissonance?