Cartesian Product of Sets

Given two non-empty sets $A$ and $B$, the Cartesian product $A \times B$ collects every possible ordered pair with its first entry from $A$ and second entry from $B$:

Order matters here — $(a, b)$ and $(b, a)$ count as different pairs unless $a = b$. If $A = \{S, M\}$ (two garment sizes) and $B = \{\text{Red}, \text{Blue}, \text{Green}\}$ (three available colours), then

representing every size-colour combination a shop would need to stock — six SKUs from two sizes and three colours. Reversing the product, $B \times A$, would instead list colour-first pairs, and $A \times B \ne B \times A$ in general.

If $A$ and $B$ are finite, every one of $A$'s elements pairs with every one of $B$'s elements exactly once, giving

For the sizes-and-colours example, $n(A \times B) = 2 \times 3 = 6$, matching the direct listing above without needing to write out a single pair. …