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# 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]a[/math] and the second coordinate is [math]b[/math] is usually written [math](a,b)[/math] (sometimes it is written [math]\langle a,b\rangle[/math]). If [math]a[/math] is different from [math]b[/math], then the ordered pair [math](a,b)[/math] is different from the ordered pair [math](b,a)[/math] - this is why it is called *ordered*.

## Properties

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

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

Kazimierz Kuratowski was the first person to make this definition.ru:Пара (математика)#Упорядоченная пара