Consider a function $f: A_f \longmapsto B_f$, where $A_f$ and $B_f$ are arbitrary sets (e.g., $A_f, B_f \subseteq \mathbb{R}$ or $A_f, B_f \subseteq \{1,2,3\}$). Consider another function $g: A_g \longmapsto B_g$, where $A_g$ and $B_g$ are again arbitrary sets. Then one can define a joint function $h$ combining the two mappings: $h = (f,g) : A_h \longmapsto B_h$ (Example: $f(x) = x$ and $g(y) = y^2$ so that $h(x,y) = (x, y^2)$).
The question is: what would be the best (i.e. common, clear, well-defined) way to define $A_h$ and $B_h$?
There seem to be at least three different possibilities:
- Tensor products: $A_h = A_f \otimes A_g$ and $B_h = B_f \otimes B_g$.
- Cartesian products: $A_h = A_f \times A_g$ and $B_h = B_f \times B_g$.
- Tuples: $A_h = (A_f, A_g)$ and $B_h = (B_f, B_g)$.
But I am not sure which should be preferred or which one might even be mandatory.
We have $h(x,y)=(f(x),g(y))$ for $(x,y) \in A_h.$
Hence $A_h$ consists of pairs $(x,y)$ such that $x \in A_f$ and $y \in A_g.$
Consequence: $A_h=A_f \times A_g$.
Furthermaore: $(f(x),g(y)) \in B_f \times B_g.$, hence $B_h = B_f \times B_g$.
Tensor products make no sense, if the sets $A_f, A_g, B_f$ and $B_g$ are arbitrary sets.
Tuples: if $M,N$ are sets, how do you define $(M,N)$ ??