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Set Notation

The Number System

The numbers used thus far in your study of mathematics can be categorized into several sets of numbers: natural, whole, integer, rational, irrational and real.

Did You Know?

In the time of Pythagoras, mathematicians thought that all numbers could be written as a ratio of two whole numbers (or, as rational numbers); the discovery of the irrational numbers necessitated a change to their known number system.

A similar development happened in the 1600s. The discovery of the complex numbers, a set that includes the square roots of negative numbers, necessitated a change to the known number system at that time. Numbers other than the real numbers existed.

In this part of the courseware, we will limit our analysis to the real numbers.

Previously you may have been introduced to these sets:

Natural numbers: $\{1,~2,~3,~4,~5, \ldots\}$: The natural numbers are the counting numbers.; The set of natural numbers is denoted by $\mathbb{N}$.; Some definitions include $0$ as a natural number; in this courseware, we will use the definition given here.
Whole Numbers: $\{0,~1,~2,~3,~4, \ldots\}$: The whole numbers are all the natural numbers together with $0$.; The set of whole numbers is denoted by $\mathbb{W}$.
Integers: $\{\ldots,~{-4},~{-3},~{-2},~{-1},~0,~1,~2,~3,~4, \ldots\}$: The integers are all the natural numbers and their negatives, as well as $0$.; The set of integers is denoted by $\mathbb{Z}$ or $\mathbb{I}$. The most common notation, and the one we will use, is $\mathbb{Z}$.
Rational numbers: The rational numbers are numbers that can be written as fractions of two integers. These are numbers whose decimal form is either terminating or periodic (repeating).; This includes all integers since integers can be written as fractions using a denominator of $1$.; The set of rational numbers is denoted by $\mathbb{Q}$.
Irrational numbers: The irrational numbers are numbers that cannot be written as a fraction of two integers. These are numbers whose decimal form is non-terminating and non-repeating.; Examples include $\pi$ and $\sqrt{5}$.; The set of irrational numbers is denoted by $\mathbb{\bar{Q}}$, a symbol that stands for "not $\mathbb{Q}$."
Real numbers: The real numbers consist of any number that is either rational or irrational.; The set of real numbers is denoted by $\mathbb{R}$.

This Venn diagram gives a visual representation of how these number sets relate.

See adjacent image description.

A Venn diagram showing that the real numbers are made up of the irrational and rational numbers. The rational numbers include all integers. Integers include all whole numbers. Whole numbers include all natural numbers.

Did You Know?

The use of the symbol "$\mathbb{Z}$" for the set of the integers seems to date back to the 1930s when it was used because of the German word "Zahlen", meaning "numbers."

The symbol "$\mathbb{Q}$" is used for the set of rational numbers because of the word "quotient", the result of a ratio of two integers.

Set Notation

A set is a collection of distinct items.

When the elements of a set can be listed, there will be no repeated elements and the order of the list does not matter. The elements of a set are listed inside braces, $\{ ~ \}$, often called 'curly brackets'.

the set of real numbers between $0$ and $1$, or
the set of rational numbers, $\mathbb{Q}$.

Set Notation Continued

How do you describe a set without listing its elements?

$\{ ~ \} ~ \rightarrow$ stands for 'The set of all...'
$\in ~\rightarrow $ stands for 'is an element of'
$\mid ~ \rightarrow$ stands for 'such that'

the set of real numbers between $0$ and $1$, and
the set of rational numbers.

Set Notation Continued

Let $A$ be the set of real numbers between $0$ and $1$, not including $0$ or $1$.

$\{ ~ \rightarrow$ 'The set of all...'
$\in ~ \rightarrow$ 'is an element of'
$\mid ~ \rightarrow$ 'such that'

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Check Your Understanding 2

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How Did I Do?Try Another

Example 3 — Part A

Doyle owns a summer company aerating lawns. He charges customers based on the size of their lawn:

Less than $650$ m²: $$25$.
From $650$ m² up to, but not including, $900$ m²: $$30$.
From $900$ m² to $1200$ m², inclusive: $$35$.
He charges a non-refundable deposit of $$25$ for any advanced bookings.

Why should the relationship between lawn size and cost be a function?

Solution — Part A

Example 3 — Part B

Doyle owns a summer company aerating lawns. He charges customers based on the size of their lawn:

Less than $650$ m²: $$25$.
From $650$ m² up to, but not including, $900$ m²: $$30$.
From $900$ m² to $1200$ m², inclusive: $$35$.
He charges a non-refundable deposit of $$25$ for any advanced bookings.

Larger equipment is needed for large lawns, so Doyle does not aerate any lawns over $1200$ m².

Graph the function.

Solution — Part B

Example 3 — Part B Continued

Solution — Part B

Non-refundable deposit of $$25$.
Less than $650$ m²: $$25$.
From $650$ m² up to, but not including, $900$ m²: $$30$.
From $900$ m² to $1200$ m², inclusive: $$35$.

Closed dot $\implies$ point is included in the function.
Open dot $\implies$ point is not included in the function.

Example 3 — Part C

Solution — Part C

Determine the domain.

Example 3 — Part D

Solution — Part D

Determine the range.

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Set Notation

The Number System

Slide Notes

Glossary

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Set Notation

Set Notation Continued

Set Notation Continued

Check Your Understanding 2

Slide Notes

Glossary

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Example 3 — Part A

Solution — Part A

Example 3 — Part B

Solution — Part B

Example 3 — Part B Continued

Solution — Part B

Example 3 — Part C

Solution — Part C

Example 3 — Part D

Solution — Part D