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In some situations, it is useful to group the objects to see the relationships in a clear way. Every object can be classified according to a characteristic. For instance, toys can be grouped according to colors. People can be grouped according to jobs or ages. In this lesson, the concept sets will be introduced especially to classify numbers.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.


Playing with Sets

The applet shows five different circles that represent number sets. Those sets are natural numbers whole numbers integers rational numbers and irrational numbers . Consider the numbers in these sets, then enlarge and arrange these circles to show the relationships between the sets.
Five Circles That Represents the Number Sets
  • Is there any set that is completely contained in another set? Think about the numbers that belong more than one number set.
  • Are there any sets that do not share any number with the other sets?

Introduction to Sets

The concept of set is used in daily life to refer simply to a group of objects. In mathematics, these objects are defined according to some properties. Now, sets will be defined based on some characteristics and the different ways to describe them.



A set is a collection of well-defined unique objects called elements. Sets are often illustrated by a Venn diagram, but there are several other ways to visualize them.

Verbal Description All negative integers greater than
Roster Notation
Set-Builder Notation

Each way of describing a set will be discussed and a corresponding example will be shown.

Verbal Description

A set can be described in words as long as there is no ambiguity of whether any particular object belongs to the set. For example, the set of delicious dishes is not a valid description because delicious is not well-defined. However, the set of desserts with chocolate is valid because including chocolate is an objective description.

Listing Method or Roster Notation

In listing method or roster notation, the elements of a set can be shown within curly brackets and separated by commas. The order of the elements is not relevant.
Furthermore, the elements of a set are only considered once and their original order can be disregarded. In the example below, a set is created with the numbers of a given list.
Although the number occurs twice in the list, it is written only once in the set .

Set-Builder Notation

In set-builder notation, sets are described by stating the common characteristics of its elements. To do this, a variable followed by or is written. Then, the common characteristics of the elements are defined. Finally, the whole notation is enclosed in curly brackets.
The symbols and are read as such that. The above example can be read as the set of all numbers such that is an odd integer greater than negative

Visualizing Sets in a Diagram

It is also possible to show a set visually by writing all the elements inside a closed shape. This method is called Venn Diagram.


Venn Diagram

A Venn diagram is a way of illustrating sets. Here, a rectangle representing the universal set, usually denoted as or is drawn. Inside the rectangle circles are typically drawn to represent sets. If the sets have elements in common, circles that represent sets must intersect. In the example, sets and are illustrated.

A venn diagram that shows sets A and B intersecting inside the universal set U.
The universal set contains all of the elements in a given collection. For example, a universal set can be all positive integer numbers that are less than Consider also two example sets and
The above sets will now be illustrated with a Venn diagram. Since the number belongs to both and it is placed in the overlapping region. Also, since the numbers and are included in neither set, they are written outside the circles but inside the rectangle that represents the universal set.
Sets A and B inside the universal set U with their elements
In this example the circles have the same size even though the sets do not have the same number of elements. This means that Venn diagrams are generally not size proportional.
Pop Quiz

Identifying the Elements in a Venn Diagram

Examine the following Venn Diagram. Determine whether the number belongs to the sets shown or list the elements of the set shown in the roster notation.

Venn Diagram Elements


If all the elements of a set are contained in another set then is a subset of This relationship can be written as A Venn diagram visualizes the expression.

A Venn diagram with Set A as a subset of Set B inside a universal set U.

As an example, consider the set All of its subsets are shown in the table.

Subset With No Elements
Subsets With One Element
Subsets With Two Elements
Subset With Three Elements
It is worth noting that all the subsets of except are called proper subsets. A proper subset of a set is a subset with fewer elements than and is denoted by
Other than that the empty set is a subset of all sets. Additionally, every set is always a subset of itself and it is denoted by This fact is illustrated with set


Another Example
Actually, sets and subsets do not necessarily include just the numbers all the time. Remember that any well-defined object can be an element of a set. For instance, consider the following sets written in verbal description.
Since all the oceans and lakes are also water bodies of the world, both the sets and are the subsets of set
Pop Quiz

Identifying the Subsets

Determine whether the given set is a subset of set or not.

Given set A and set B

Number Sets

A number set is a collection of numbers that allows different types of numbers to be placed in various categories. Listed are some of the most common number sets.

  • Natural Numbers, also called counting numbers, are the numbers used for counting. The set of natural numbers is denoted by
  • Whole Numbers are the natural numbers in addition to zero. The set of whole numbers is denoted by
  • Integer Numbers are the whole numbers in addition to their opposites. The set of integer numbers is denoted by
  • Rational Numbers are the numbers that can be expressed as a ratio between two integers. The set of rational numbers is denoted by
  • Irrational Numbers are the numbers that cannot be expressed as the ratio between two integers. The set of irrational numbers can be written as
  • Real Numbers include rational numbers in addition to irrational numbers. The set of real numbers is denoted by

Notice that the elements of some numbers sets are also the elements of another number set. For instance, all natural numbers are also integers. This means that natural numbers are the subset of integers. With these in mind, consider the applet at the beginning of the lesson once again. The relations between the number sets can be shown as the following way.

Number Sets in Venn Diagram

In the following table, each set will be shown by using the roster notation.

Number Set Roster Notation
Natural Numbers
Whole Numbers
Integer Numbers
Rational Numbers
Irrational Numbers
Real Numbers

Real Numbers

The combination of rational and irrational numbers makes the set of real numbers represented by Represented by a graph, the set of real numbers is formed by continuous values marked on a number line.
Real numbers on a number line
Real numbers are the numbers that can be found in the real world. For instance, natural numbers are used for counting objects, rational numbers are used for representing fractions, irrational numbers are used for calculating the square root of a number, integers for measuring temperature, and so on. These different types of numbers make a collection of real numbers.
Pop Quiz

Identifying the Sets of Numbers

Recall that a number can belong to more than one number set. For instance, is a whole number, but it is an integer, a rational number, and a real number at the same time. Consider the given numbers in the following applet and determine all the number sets that include the number.

Random Numbers

Sorting the Trophies

Vincenzo is a professional athlete and captain of the an amputee football team in San Francisco — Bayside Flyers. Since he became the team captain, the Flyers have won five different trophies. Vincenzo wants to display the trophies in order of height from shortest to tallest.

A boy in a wheelchair sits on the bottom left. In front of him, there is a table with five trophies. Above him, there is a clock, and above the trophies is a picture.
The heights of the trophies are    and  inches. Help him to sort these heights from shortest to tallest.


Round all the numbers to one decimal place and plot them on a number line.


The heights of the trophies are given as    and  inches. These numbers will be rewritten as decimal numbers and rounded to one decimal place to write them in ascending order. Note that is already in decimal form. Start by expanding
Now round it to the nearest tenth.
The next step is to write as a decimal. Recall that is equal to
Now, approximate by using perfect squares. The two nearest perfect squares are and
Try to find a better approximation by using decimals. Choose several decimals bigger than and less than Then, calculate the square of each number and compare them with
Approximation Square of Approximation Comparison
Approximation is low
Approximation is low
Approximation is low
Approximation is low
Approximation is high
Considering the squares of approximations, is closer to This means that is closer to Now, calculate the square of
Finally, mark all the approximations on a number line.
Number Line
The order of the rounded numbers in decimal form can be seen from the number line. Now Vincenzo can properly sort the heights of the trophies from shortest to tallest.

Beyond Real Numbers

In this lesson, it was mentioned that natural numbers, whole numbers, integers, rational numbers, and irrational numbers are real numbers. Considering that vast range, the set of real numbers must be the biggest number set ever. Right?🤔

Numbers that are not real.

Well, there is what is called an imaginary number Imaginary numbers combined with real numbers form a set of complex numbers

Number Sets in Venn Diagram
These number sets will be explored in later courses.