How does the Cartesian product of two sets create ordered pairs?

In set theory, the Cartesian product of two sets, denoted as $A\times B$, creates a new set by pairing every element from the first set with every element from the second set.1

Specifically, the Cartesian product consists of all ordered pairs $(a,b)$ such that the first element $a$ is a member of set $A$ and the second element $b$ is a member of set $B$. Unlike a standard set where the order of elements does not matter, an ordered pair is defined by the specific sequence of its components; therefore, $(a,b)$ is generally not equal to $(b,a)$ unless $a=b$.21

For example, if set $A=\{x,y\}$ and set $B=\{3,6,9\}$, the Cartesian product $A\times B$ results in the set of pairs: $\{(x,3),(x,6),(x,9),(y,3),(y,6),(y,9)\}$. This construction is fundamental to analytic geometry, where it is used to define the coordinates of points on a grid.1