Definition:Ordered Tuple

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Definition

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

Let $\N^*_n$ be the first $n$ non-zero natural numbers:

$\N^*_n := \set {1, 2, \ldots, n}$


Definition 1

An ordered tuple (of length $n$) is a finite sequence whose domain is $\N^*_n$.


Definition 2

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


Ordered Tuple on Set

Let $S$ be a set.

Let $s: \N^*_n \to S$ be an ordered tuple.


Then $s$ is called an ordered tuple on $S$, its codomain.


Empty Ordered Tuple

Let $S$ be a set.

The empty ordered tuple on $S$ is the empty mapping:

$\O \to S$

from the empty set $\O$ to $S$.


Ordered Tuple Defined by Sequence

Let $\sequence {a_k}_{k \mathop \in A}$ be a finite sequence of $n$ terms.

Let $\sigma$ be a permutation of $A$.


Then the ordered $n$-tuple defined by the sequence $\sequence {a_{\map \sigma k} }_{k \mathop \in A}$ is the ordered $n$-tuple:

$\sequence {a_{\map \sigma {\map \tau j} } }_{1 \mathop \le j \mathop \le n}$

where $\tau$ is the unique isomorphism from the totally ordered set $\closedint 1 n$ onto the totally ordered set $A$.


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 defined as

Some treatments take the intuitive approach of regarding an ordered tuple merely as an ordered set, that is, without stressing the fact of it being a mapping from a subset of the natural numbers:

Ordered Tuple as Ordered Set

The ordered tuple $\tuple {a_1, a_2, \ldots, a_n}$ of elements $a_1, a_2, \ldots, a_n$ is defined as either the ordered pair:

$\tuple {a_1, \tuple {a_2, a_3, \ldots, a_n} }$

or as the ordered pair:

$\tuple {\tuple {a_1, a_2, \ldots, a_{n - 1} }, a_n}$

where $\tuple {a_2, a_3, \ldots, a_n}$ and $\tuple {a_1, a_2, \ldots, a_{n - 1} }$ are themselves ordered tuples.


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 $\tuple {x_1, x_2, \dotsc, x_n}$.


Ordered Couples and Ordered Pairs

Notice the difference between ordered pairs and ordered couples.

By definition, an ordered couple $\tuple {a, b}$ is in fact the set $\set {\tuple {1, a}, \tuple {2, b} }$, where each of $\tuple {1, a}$ and $\tuple {2, b}$ are ordered pairs.


It is not possible to use the definition of ordered couple as the definition of ordered pair, as the latter is used to define a mapping, which is then used to define an ordered couple.

However, in view of the Equality of Ordered Tuples, it is generally accepted that it is valid to use the notation $\tuple {a, b}$ to mean both an ordered couple and an ordered pair.

It is worth bearing this in mind, as there are times when it is important not to confuse them.


Also see

  • Results about ordered tuples can be found here.


Sources

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