# Sets of Ordered Pairs

## Prerequisites

### Sets of Ordered Pairs

#### Introduction

• We have special symbols denoting the set of all integers and the set of all real numbers: Z and R respectively.
• These can be modified to denote the set of all ordered pairs:
• Z2 denotes the set of all ordered pairs of integers.
• R2 denotes the set of all ordered pairs of real numbers.

#### The set of all ordered pairs of integers

• Z2 contains every possible pair of integers, e.g.
• We can define Z2 in terms of Z using set builder notation:
LaTeX:\mathbb{Z}^2 = \{ (z_1, z_2)\colon z_1, z_2 \in \mathbb{Z} \}

#### The set of all ordered pairs of real numbers

• R2 contains every possible pair of real numbers, e.g.
• We can define R2 in terms of R using set builder notation:
LaTeX:\mathbb{R}^2 = \{ (r_1, r_2)\colon r_1, r_2 \in \mathbb{R} \}

#### Higher dimensions

• R and Z can also be extended to higher dimensions:
• To represent the set of all triplets of integers, you can use a superscript of 3: