Definition:Ordered n-Tuple

Definition

An ordered $n$-tuple is a finite sequence whose domain is $\N^*_n$.

If $\left \langle {a_k} \right \rangle_{k \in \N^*_n}$ is an ordered $n$-tuple, then $a_k$ is called the $k$th term of the ordered $n$-tuple for each $k \in \N^*_n$.

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

An ordered $n$-tuple can (and often will) be denoted $\left({a_1, a_2, \ldots, a_n}\right)$ instead of by $\left \langle {a_k} \right \rangle_{1 \le k \le n}$ etc.

As an example, $\left({6, 3, 3}\right)$ is the ordered triple $f$ defined by $f \left({1}\right) = 6, f \left({2}\right) = 3, f \left({3}\right) = 3$.

In order to further streamline notation, it is common to use the more compact $\left \langle {a_n} \right \rangle$ for $\left \langle {a_k} \right \rangle_{1 \le k \le n}$.

It will be understood that the subscript runs through all values of $\N^*_n$ from $1$ to $n$.

Ordered n-tuple Defined by a Sequence

Let $\left \langle {a_k} \right \rangle_{k \in A}$ be a finite sequence of $n$ terms.

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

Then the ordered $n$-tuple defined by the sequence $\left \langle {a_{\sigma \left({k}\right)}} \right \rangle_{k \in A}$ is the ordered $n$-tuple:

$\left \langle {a_{\sigma \left({\tau \left({j}\right)}\right)}}\right \rangle_{1 \le j \le n}$

where $\tau$ is the unique isomorphism from the totally ordered set $\left[{1 \,.\,.\, n}\right]$ onto the totally ordered set $A$.

Equality of Ordered n-Tuples

Let:

$(1): \quad \left \langle {a_m} \right \rangle = \left({a_1, a_2, \ldots, a_m}\right)$

and

$(2): \quad \left \langle {b_n} \right \rangle = \left({b_1, b_2, \ldots, b_n}\right)$

be ordered tuples for some $m, n \in \N^*$. Then:

$\left \langle {a_m} \right \rangle = \left \langle {b_n} \right \rangle \iff n = m \land \forall j \in \N^*_n: a_j = b_j$

Comment

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

Notice the difference between ordered pairs and ordered couples.

By definition, an ordered couple $\left({a, b}\right)$ is in fact the set $\left\{{\left({1, a}\right), \left({2, b}\right)}\right\}$, where each of $\left({1, a}\right)$ and $\left({2, b}\right)$ 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 $\left({a, b}\right)$ 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.

Alternative Notation

Various alternatives to $\left({a_1, a_2, \ldots, a_n}\right)$ can be found in the literature, for example:

$\left \langle {a_1, a_2, \ldots, a_n} \right \rangle$

This notation is recommended when use of parentheses would be ambiguous.

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

Other notations which may be encountered are:

$\left[{a_1, a_2, \ldots, a_n}\right]$

$\left\{{a_1, a_2, \ldots, a_n}\right\}$

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.

Also see

• Ordered Tuple as Ordered Set

Sources

• Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts (1951): Introduction $\S 2$
• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 9$: Families
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 18$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 0.4$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 6$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 8$
• H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): Appendix $\text{A}.10$