Tuples as nested ordered pairs

Another way of formalizing tuples is as nested ordered pairs:

1. The 0-tuple (i.e. the empty tuple) is represented by the empty set .

2. An -tuple, with , can be defined as an ordered pair of its first entry and an -tuple (which contains the remaining entries when ):

Thus, for example:

In mathematics, a Cartesian product is a mathematical operation which returns a set (or product set) from multiple sets. The Cartesian product is the result of crossing members of each set with one another.

The simplest case of a Cartesian product is the Cartesian square, which returns a set from two sets. A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product rows ×columns is taken, the cells of the table contain ordered pairs of the form (row value,column value). If the Cartesian product is columns × rows is taken, the cells of the table contain the ordered pairs of the form (column value,row value).

A Cartesian product of n sets can be represented by an array of n dimensions, where each element is an n-tuple. An ordered pair is a 2-tuple.

The Cartesian product is named after René Descartes,^[1] whose formulation of analytic geometry gave rise to the concept.

3.A correspondence is an ordered triple (X,Y,R), where R is a relation from X to Y, i.e. any subset of the Cartesian product X×Y. In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function. In graph theory, a map is a drawing of a graph on a surface without overlapping edges (a planar graph), similar to a political map.

In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments and images are related ormapped to each other.

A function maps elements from its domain to elements in its codomain.

A function is injective (one-to-one) if every element of the codomain is mapped to by at most one element of the domain. Notationally,

or, equivalently (using logical transposition),

An injective function is an injection.

A function is surjective (onto) if every element of the codomain is mapped to by at least one element of the domain. (That is, the image and the codomain of the function are equal.) Notationally,

A surjective function is a surjection.

A function is bijective (one-to-one and onto or one-to-one correspondence) if every element of the codomain is mapped to by exactly one element of the domain. (That is, the function is both injective and surjective.) A bijective function is a bijection.

An injective function need not be surjective (not all elements of the codomain may be associated with arguments), and a surjective function need not be injective (some images may be associated with more than one argument). The four possible combinations of injective and surjective features are illustrated in the following diagrams.

Injective and surjective (bijection)	Injective and non-surjective (injection, or one-to-one)
Non-injective and surjective (surjection, or onto)	Non-injective and non-surjective (projection)