Ordered pair

Introduction

In mathematics, an ordered pair is a collection of two objects, where one of the objects is first (the first coordinate or left projection), and the other is second (the second coordinate or right projection). An ordered pair where the first coordinate is [math]\displaystyle{ a }[/math] and the second coordinate is [math]\displaystyle{ b }[/math] is usually written [math]\displaystyle{ (a,b) }[/math] (sometimes it is written [math]\displaystyle{ \langle a,b\rangle }[/math]). If [math]\displaystyle{ a }[/math] is different from [math]\displaystyle{ b }[/math], then the ordered pair [math]\displaystyle{ (a,b) }[/math] is different from the ordered pair [math]\displaystyle{ (b,a) }[/math] - this is why it is called ordered.

Properties

If [math]\displaystyle{ (a_1,b_1) }[/math] and [math]\displaystyle{ (a_2,b_2) }[/math] are two ordered pairs, then the characteristic or defining property of ordered pairs is:

[math]\displaystyle{ (a_1,b_1) = (a_2, b_2) \leftrightarrow a_1 = a_2 \and b_1 = b_2 }[/math].

This means that two ordered pairs are equal if and only if: the first coordinates of the pairs are equal, and also the second coordinates of the pairs are equal.

Definition

There are many mathematical definitions of ordered pair which have this property. The definition given here is the most common one:

[math]\displaystyle{ (a,b) = \{\{a\}, \{a,b\}\} }[/math].

Kazimierz Kuratowski was the first person to make this definition.