Sets of Ordered Pairs

Prerequisites

Introduction to Sets Show

Ordered Pairs Show

Important Sets Show

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: