# Definition:Lexicographic Order/Tuples of Equal Length/Cartesian Space

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## Contents

## Definition

Let $\left({S, \preceq}\right)$ be an ordered set.

Let $n \in \N_{>0}$.

Let $S^n$ be the cartesian $n$th power of $S$:

- $S^n = \underbrace{S \times S \times \cdots \times S}_{n \text{ times} }$

The **lexicographic order on $S^n$** is the relation $\preccurlyeq$ defined on $S^n$ as:

- $\left({x_1, x_2, \ldots, x_n}\right) \preccurlyeq \left({y_1, y_2, \ldots, y_n}\right)$ if and only if:
- $\exists k: 1 \le k \le n: \left({\forall j: 1 \le j < k: x_j = y_j}\right) \land \left({x_k \prec y_k}\right)$

- or:
- $\forall j: 1 \le j \le n: x_j = y_j$

That is, if and only if:

- the elements of a pair of $n$-tuples are either all equal

or:

- they are all equal up to a certain point, and on the next one they are comparable and they are different.

## Also known as

**Lexicographic order** can also be known as the more unwieldy **lexicographical ordering**.

## Also see

- Results about
**Lexicographic Order**can be found here.

## Sources

- 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.1$: Mathematical Induction: Exercise $15 \ \text{(d)}$