Cartesian Product of Two Sets

Given two non-empty sets $A$ and $B$, the Cartesian product $A \times B$ is the set of all possible ordered pairs $(a, b)$ where the first entry $a$ comes from $A$ and the second entry $b$ comes from $B$:

The word ordered matters — $(a, b)$ and $(b, a)$ are treated as different pairs unless $a = b$. For example, if $A = \{1, 2\}$ and $B = \{3, 4\}$, then

Notice $A \times B \ne B \times A$ here — the Cartesian product is not commutative in general.

Every element of $A$ is paired with every element of $B$ exactly once, so if $A$ has $m$ elements and $B$ has $n$ elements, $A \times B$ has exactly $m \times n$ ordered pairs — this connects directly to the idea of cardinality covered next. …