Definition:Cartesian Product/Coordinate
Definition
Let $\ds \prod_{i \mathop \in I} S_i$ be a cartesian product.
Let $j \in I$, and let $s = \sequence {s_i}_{i \mathop \in I} \in \ds \prod_{i \mathop \in I} S_i$.
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, for example, the first, second, or $n$th coordinate.
For an element $\tuple {s, t} \in S \times T$ of a binary cartesian product, $s$ is the first coordinate, and $t$ is the second coordinate.
Also denoted as
It is usual to use the subscript technique to denote the coordinates where $n$ is large or unspecified:
- $\tuple {x_1, x_2, \ldots, x_n}$
However, note that some texts (often in the fields of physics and mechanics) prefer to use superscripts:
- $\tuple {x^1, x^2, \ldots, x^n}$
While this notation is documented here, its use is not endorsed by $\mathsf{Pr} \infty \mathsf{fWiki}$ because:
- there exists the all too likely subsequent confusion with notation for powers
- one of the philosophical tenets of $\mathsf{Pr} \infty \mathsf{fWiki}$ is to present a system of notation that is as completely consistent as possible.
Linguistic Note
It's an awkward word coordinate.
It really needs a hyphen in it to emphasise its pronunciation (loosely and commonly: coe-wordinate), and indeed, some authors spell it co-ordinate.
However, this makes it look unwieldy.
An older spelling puts a diaeresis indication symbol on the second "o": coördinate.
But this is considered archaic nowadays and few sources still use it.
Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 9$
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 5$: Cartesian Products
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Order