Definition:Cartesian Product/Coordinate

Definition
Let $\displaystyle \prod_{i \mathop \in I} S_i$ be a cartesian product.

Let $j \in I$, and let $s = \left({s_i}\right)_{i \in I} \in \displaystyle \prod_{i \mathop \in I} S_i$.

Then $s_j$ is called the $j$th coordinate of $s$.

If the indexing set $I$ consists of ordinary numbers $1, 2, \ldots, n$, one speaks about e.g. the first, second, or $n$th coordinate.

For an element $\left({s, t}\right) \in S \times T$ of a binary cartesian product, $s$ is the first coordinate, and $t$ is the second coordinate.