Ordered Pairs and Cartesian Product

So far the order of elements inside a set did not matter. But many real situations pair two things in a definite order — a product with its price, a year with its sales figure, an $x$-coordinate with a $y$-coordinate. For these we need the idea of an ordered pair.

Ordered pair. An ordered pair $(a, b)$ is a pairing of two objects in which $a$ is the first component and $b$ the second. Unlike a set, order is essential:

So $(2, 3) \ne (3, 2)$, whereas the sets $\{2, 3\}$ and $\{3, 2\}$ are equal. This is exactly why a point in the plane is written as an ordered pair of coordinates.

Cartesian product. Given two non-empty sets $A$ and $B$, the Cartesian product $A \times B$ (read "$A$ cross $B$") is the set of all ordered pairs whose first component comes from $A$ and second component from $B$:

For example, if $A = \{1, 2\}$ and $B = \{x, y\}$, then

Counting the pairs. Since each of the $n(A)$ choices for the first component can be paired with each of the $n(B)$ choices for the second: