Name of $\{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)\}$ in terms of $\{1,2,3\}$

Question

This question is more about nomenclature than a problem to solve.

Imagine that I have a set

$$A = \{1,2,3\}$$

and I want do make the following set

$$B = \{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)\}$$

How would I call the set $B$? Is this the set of all permutated pairs with all elements of $A$? But is this even permutations or combinations? How can I call $B$ in terms of $A$?

Answer 1

We can write this as $B = A\times A$, where $\times$ denotes the cartesian product. Alternatively, you might write $A^2$.

In words, you might say that $B$ is the set of all ordered pairs of (elements of) $A$.