Operation producing a set of ordered pairs from a pair of ordered sets

Question

Consider two indexed sets $A=\left(a_{1}, a_{2}, \dots, a_{n}\right)$ and $B=\left(b_{1}, b_{2}, \dots, b_{n}\right)$. From a mathematical standpoint, what would be the name of the operation $\odot$ such that $$C = A \odot B = \left\{\left(a_{1}, b_{1}\right), \left(a_{2}, b_{2}\right), \dots, \left(a_{n}, b_{n}\right)\right\}$$ This is a very common operation in programming, but I suddenly cannot see if it corresponds to a common operation on sets/sequences.

Answer 1

An "indexed set" is really a function whose domain is some "indexing set" $I$. For instance, what you write as $A=(a_1,\dots,a_n)$ is actually a function $f$ on the domain $\{1,\dots,n\}$ with $f(i)=a_i$.

When you consider two indexed sets with the same indexing set $I$ as functions $f$ and $g$, the standard notation for the construction you describe is $(f,g)$. That is, $(f,g)$ denotes the function on $I$ given by $(f,g)(i)=(f(i),g(i))$. (Strictly speaking this is of course ambiguous, since $(f,g)$ could also just denote the ordered pair of $f$ and $g$, but in practice the meaning is usually clear.)

I don't know of any especially commonly used name for this construction. Some names for $(f,g)$ you might encounter in a category-theoretic context are the "pairing" or "tupling" (or sometimes "product") of $f$ and $g$.