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Here are a few recommended readings before getting started with this lesson.
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 $-5.$ |
---|---|
Roster Notation | ${-1,-2,-3,-4}$ |
Set-Builder Notation | ${x∣xis a negative integer greater than-5}$ |
Each way of describing a set will be discussed and a corresponding example will be shown.
the set of delicious dishesis not a valid description because delicious is not well-defined. However,
the set of desserts with chocolateis valid because including chocolate is an objective description.
$∣$or
$:$is written. Then, the common characteristics of the elements are defined. Finally, the whole notation is enclosed in curly brackets.
$∣$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 $3.$
It is also possible to show a set visually by writing all the elements inside a closed shape. This method is called 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.
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.
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 | ${1,2,3}$ |
$⊂.$
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.
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 | ${…,-23 ,-2,-1,0,1,2,95 ,…}$ |
Irrational Numbers | ${π,e,2 ,…}$ |
Real Numbers | ${…,-23 ,-2,-1,0,1,2,95 ,π,e,2 ,…}$ |
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.
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.
The heights of the trophies are $π_{2},$ $821 ,$ $90 ,$ $8.6,$ and $8.6$ 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.
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 |
Rewrite $π$ as $3.141592…$
Multiply
Round to $1$ decimal place(s)
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 $C.$
These number sets will be explored in later courses.