Definition:Ordered Pair

The definition of a set does not take any account of the order in which the elements are listed.

That is, $$\left\{{a, b}\right\} = \left\{{b, a}\right\}$$, and the elements $$a$$ and $$b$$ have the same status - neither is distinguished above the other as being more "important".

An ordered pair is a two-element set together with an ordering.

In other words, one of the elements is distinguished above the other - it comes first.

Such a structure is written:
 * $$\left({a, b}\right)$$

and it means: "first $$a$$, then $$b$$".

It can be formalised by the definition:


 * $$\left({a, b}\right) = \left\{{\left\{{a}\right\}, \left\{{a, b}\right\}}\right\}$$

In an ordered pair $$\left({a, b}\right)$$, the following terminology is used:
 * $$a$$ is called the first coordinate;
 * $$b$$ is called the second coordinate.

This definition is compatible with the equivalent definition in the context of Cartesian coordinates.