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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.
Explore
Playing with Sets
The applet shows five different circles that represent number sets. Those sets are natural numbers N, whole numbers W, integers Z, rational numbers Q, and irrational numbers R\ Q. Consider the numbers in these sets, then enlarge and arrange these circles to show the relationships between the 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?
Discussion
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.
Concept
Set
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 - 5. |
|---|---|
| Roster Notation | {-1, -2, -3, -4} |
| Set-Builder Notation | { x | x is a negative integer greater than- 5 } |
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.
Example Set I:& Desserts with chocolate Example Set II:& Capital cities of the world Example Set III:& European countries
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. Verbal Description:& Factors of9 Roster Notation:& {1,3,9} 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. List:3, 3, 1, 2 ↓ Set:{1,2,3} Although the number 3 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.
{ -3,-1,1,3,5,... } ↓ { x | x is an odd integer greater than- 3 }
|and
:are read as
such that.The above example can be read as
the set of all numbers x such that x is an odd integer greater than negative 3.
Discussion
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.
Concept
Venn Diagram
A Venn diagram is a way of illustrating sets. Here, a rectangle representing the universal set, usually denoted as U
or E,
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 A and B are illustrated.
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.
Discussion
Subset
If all the elements of a set A are contained in another set B, then A is a subset of B. This relationship can be written as A⊆ B.
A Venn diagram visualizes the expression.
As an example, consider the set B={1,2,3}. All of its subsets are shown in the table.
| Subset With No Elements | { } or ∅ |
|---|---|
| Subsets With One Element | { 1 },{ 2 },{ 3 } |
| Subsets With Two Elements | { 1,2 },{ 2,3 },{ 1,3 } |
| Subset With Three Elements | { 1,2,3 } |
It is worth noting that all the subsets of B except { 1,2,3 } are called proper subsets. A proper subset of a set B is a subset with fewer elements than B and is denoted by ⊂.
{1,2} ⊂ B
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 B.
{ } ⊂ B { 1,2,3 } ⊆ B
Extra
Another ExampleActually, 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. Set A:&Water bodies of the world Set B:&Oceans Set C:&Lakes Since all the oceans and lakes are also water bodies of the world, both the sets B and C are the subsets of set A.
Pop Quiz
Identifying the Subsets
Discussion
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 N.
- Whole Numbers are the natural numbers in addition to zero. The set of whole numbers is denoted by W.
- Integer Numbers are the whole numbers in addition to their opposites. The set of integer numbers is denoted by Z.
- Rational Numbers are the numbers that can be expressed as a ratio between two integers. The set of rational numbers is denoted by Q.
- Irrational Numbers are the numbers that cannot be expressed as the ratio between two integers. The set of irrational numbers can be written as R\ Q.
- Real Numbers include rational numbers in addition to irrational numbers. The set of real numbers is denoted by R.
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.
In the following table, each set will be shown by using the roster notation.
| Number Set | Roster Notation |
|---|---|
| Natural Numbers | {1,2,3, ... } |
| Whole Numbers | {0,1,2,3, ... } |
| Integer Numbers | {...,-2,-1,0,1,2, ... } |
| Rational Numbers | {...,-3/2,-2,-1,0,1,2,5/9, ... } |
| Irrational Numbers | { π,e,sqrt(2),...} |
| Real Numbers | {...,-3/2,-2,-1,0,1,2,5/9,π,e,sqrt(2), ... } |
Discussion
Real Numbers
The combination of rational and irrational numbers makes the set of real numbers represented by R. Represented by a graph, the set of real numbers is formed by continuous values marked 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. π , e , sqrt(2) , 3/5, -4, 22
Pop Quiz
Identifying the Sets of Numbers
Recall that a number can belong to more than one number set. For instance, 0 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.
Example
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.
{"type":"sort","form":{"alts":[{"id":0,"text":"8 12"},{"id":1,"text":"8.6"},{"id":2,"text":"8.6"},{"id":3,"text":"sqrt(90)"},{"id":4,"text":"\u03c0^2"}]},"formTextBefore":"","formTextAfter":"","answer":[0,1,2,3,4]}
Hint
Round all the numbers to one decimal place and plot them on a number line.
Solution
The heights of the trophies are given as π^2, 8 12, sqrt(90), 8.6, and 8.6 inches. These numbers will be rewritten as decimal numbers and rounded to one decimal place to write them in ascending order. Note that 8.6 is already in decimal form. Start by expanding 8.6. 8.6 = 8.6666...
Now round it to the nearest tenth.
8.6 ≈ 8.7
The next step is to write 8 12 as a decimal. Recall that 12 is equal to 0.5.
8 12 = 8.5
Now, approximate sqrt(90) by using perfect squares. The two nearest perfect squares are 81 and 100.
Try to find a better approximation by using decimals. Choose several decimals bigger than 9 and less than 10. Then, calculate the square of each number and compare them with 90.
Considering the squares of approximations, 90 is closer to 90.25. This means that sqrt(90) is closer to 9.5. Now, calculate the square of π.
Finally, mark all the approximations on a number line. 
| Approximation | Square of Approximation | Comparison |
|---|---|---|
| 9.1 | 9.1 * 9.1 = 82.81 | Approximation is low |
| 9.2 | 9.2 * 9.2 = 84.64 | Approximation is low |
| 9.3 | 9.3 * 9.3 = 86.49 | Approximation is low |
| 9.4 | 9.4 * 9.4 = 88.36 | Approximation is low |
| 9.5 | 9.5 * 9.5 = 90.25 | Approximation is high |
π^2 =π * π
Rewrite
Rewrite π as 3.141592...
π^2 =(3.141592...) * (3.141592...)
Multiply
Multiply
π^2=9.869604...
RoundDec
Round to 1 decimal place(s)
π^2 ≈ 9.9
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. 8.5 & < & 8.6 & < & 8.7 & < & 9.5 & < & 9.9 8 12 & < & 8.6 & < & 8.6 & < & sqrt(90) & < & π^2
Closure
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?🤔
Well, there is what is called an imaginary number I. Imaginary numbers combined with real numbers form a set of complex numbers C.
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