# Definition:Ordered Tuple/Definition 2

## Definition

Let $n \in \N$ be a natural number.

Let $\N_n$ denote the first $n$ non-zero natural numbers:

- $\N_n := \set {1, 2, \ldots, n}$

Let $\family {S_i}_{i \mathop \in \N_n}$ be a family of sets indexed by $\N_n$.

Let $\displaystyle \prod_{i \mathop \in \N_n} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in \N_n}$.

An **ordered tuple of length $n$** of $\family {S_i}$ is an element of $\displaystyle \prod_{i \mathop \in \N_n} S_i$.

## Term of Ordered Tuple

Let $\sequence {a_k}_{k \mathop \in \N^*_n}$ be an ordered tuple.

The ordered pair $\tuple {k, a_k}$ is called the **$k$th term** of the ordered tuple for each $k \in \N^*_n$.

## Also known as

Some sources refer to an ordered tuple as a **tuple**.

The term **ordered $n$-tuple** can sometimes be seen, particularly for specific instances of $n$.

Instead of writing **2-tuple**, **3-tuple** and **4-tuple**, the terms **couple**, **triple** and **quadruple** are usually used.

In the context of abstract algebra, the concept is encountered as **(associative) word**.

## Notation

Notation for an ordered tuple varies throughout the literature.

There are also specialised instances of an ordered tuple where the convention is to use angle brackets.

However, it is common for an ordered tuple to be denoted:

- $\tuple {a_1, a_2, \ldots, a_n}$

extending the notation for an ordered pair.

For example: $\tuple {6, 3, 3}$ is the ordered triple $f$ defined as:

- $\map f 1 = 6, \map f 2 = 3, \map f 3 = 3$

The notation:

- $\sequence {a_1, a_2, \ldots, a_n}$

is recommended when use of round brackets would be ambiguous.

Other notations which may be encountered are:

- $\sqbrk {a_1, a_2, \ldots, a_n}$
- $\set {a_1, a_2, \ldots, a_n}$

but both of these are strongly discouraged: the square bracket format because there are rendering problems on this site, the latter because it is too easily confused with set notation.

In order to further streamline notation, it is common to use the more compact $\sequence {a_n}$ for $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$.

Some sources, particularly in such fields as communication theory, where the elements of the domain of the ordered tuple is a specific set of symbols, use the notation $x_1 x_2 \cdots x_n$ for $\tuple {x_1, x_2, \dotsc, x_n}$.

## Also see

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 9$: Families - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.1$: Functions