Important Sets

Prerequisites

Introduction to Sets Show

Important Sets

Introduction

  • Some sets are used so often in mathematics that they are given their own names.
  • This article will cover two of them:
    • Z: The set of all integers.
    • R: The set of all real numbers.

Integers

  • The set of all integers contains all the whole numbers, including negative numbers and 0.
  • It is represented by an uppercase bold (or blackboard bold) Z:
    LaTeX:\mathbb{Z} = \{ ..., -3, -2, -1, 0, 1, 2, 3, ... \}

code (Python)

i = 1
print(type(i))  # prints <class 'int'>

Real Numbers

  • The set of all real numbers contains every number that is not imaginary. This includes:
    • Integers: e.g. 0, 1, 2
    • Rational Numbers: Those which can be represented as a fraction, e.g. 0.5, 1.1
    • Irrational Numbers: Those which cannot be represented by a fraction, e.g.
  • It is represented by an uppercase bold (or blackboard bold) R:
    LaTeX:\mathbb{R} = \{ -1, 1.4, \pi \}

code (Python)

Unfortunately we cannot represent irrational numbers using code. Instead we can approximate them as floating point numbers:
r = 1.41421356237
print(type(r))  # prints <class 'float'>