# 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 $\ds \prod_{i \mathop \in \N_n} S_i$ 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 $\ds \prod_{i \mathop \in \N_n} S_i$.

## 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** or just **$n$-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.

However, beware of **couple**, as this has a completely different meaning in the context of mechanics.

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

It is noted that an **ordered tuple** is in fact a one-dimensional array.

## 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

- Results about
**ordered tuples**can be found**here**.

## 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