Definition – Tuple. – Math-reference.net

(!) Ordered pair also links here.
(!) Article is incomplete.

Definition

A tuple in Mathematics is a finite set of objects, written as $(a_1, a_2, ..., a_n)$ . We define them in terms of sets as follows.

First, we define $(a_0) := () := \emptyset$
Next, we define an ordered pair as follows: $(a_{i-1}, a_i) = \{a_{i-1}, \{a_{i-1}, a_i\}\}$ .
Finally, we define a tuple of $n$ elements to be the following set using a recurrence relation:

$(a_1, a_2, ..., a_n) = \{(a_1, a_2, ..., a_{n-1}), \{(a_1, a_2, ..., a_{n-1}), a_n\},~~~$

Where $epsilon > 0$ is a real number.

Notes

This definition is analogous to the definition of a $epsilon$ -Neighborhood of $x$ , except that we do not include the point $x$ itself. This is indicated by the fact that $|x - y| < 0$ . Since $|x - y| neq 0$ (because we haven’t written $geq$ ), it follows that $y neq x$ .