1.5: Introduction to Sets and Real Numbers

Sets - An Introduction

A set is a collection of objects. The objects in a set are called its elements or members. The elements in a set can be any types of objects, including sets! The members of a set do not even have to be of the same type. For example, although it may not have any meaningful application, a set can consist of numbers and names.

We usually use capital letters such as \(A\), \(B\), \(C\), \(S\), and \(T\) to represent sets, and denote their generic elements by their corresponding lowercase letters \(a\), \(b\), \(c\), \(s\), and \(t\), respectively. To indicate that \(b\) is an element of the set \(B\), we adopt the notation \(b\in B\), which means “ \(b\) belongs to \(B\)” or “ \(b\) is an element of \(B\).”  Consequently, saying \(x\in\mathbb {R}\) is another way of saying \(x\) is a real number.

Definition: Subset

Set A is a subset of Set B if and only if every element in Set A is also in Set B.

In symbols:

\[A \subset B \iff x\in A \rightarrow x\in B\]

Real Numbers and some Subsets of Real Numbers

We designate these notations for some special sets of numbers: \[\begin{aligned} \mathbb{N} &=& \mbox{the set of natural numbers}, \\ \mathbb{Z} &=& \mbox{the set of integers}, \\ \mathbb{Q} &=& \mbox{the set of rational numbers},\\ \mathbb{R} &=& \mbox{the set of real numbers}. \end{aligned}\] All these are infinite sets, because they all contain infinitely many elements. In contrast, finite sets contain finitely many elements. 

 

We list the natural numbers and integers while defining the rational, real and irrational numbers.

 

\(\mathbb{N} = \{1, 2, 3,\ldots \}\)

\(\mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3,\ldots \}\)

 

Definition - Rational Numbers

A rational number is a number that can be expressed as a ratio of two integers (with the second integer not equal to zero). Hence, a rational number can be written as \(\frac{m}{n}\) for some integers \(m\) and \(n\), where \(n\neq0\).

 

Definition - Real Numbers

The real numbers are the numbers corresponding to all the points on the number line.

 

Definition - Irrational Numbers

An irrational number is a real number that can not be expressed as a ratio of two integers; i.e., is not rational.

 

Closure

Definition

Given a set S with a binary operation *, S is closed under the operation * if and only if \(x*y \in S \mbox{ for every }x \in S\mbox{ and for every } y\in S\).

Example \(\PageIndex{1}\)

Suppose you add any two integers together. Will the sum always be an integer?

Solution

Yes; that's why the set of integers is closed under addition.

 

Assumption

We will use the property that the set of integers is closed under addition, subtraction and multiplication.

Alternate syntax is "closure of integers under multiplication".

This assumption can be used as a reason in an explanation or a proof.

Example \(\PageIndex{2}\)

If \(a,b \in \mathbb{Z}\text{, then }a+b \in \mathbb{Z}\) because ?

Solution

The set of integers is closed under addition.

Set Notation

Roster Notation

We can use the roster notation to describe a set if it has only a small number of elements. We list all its elements explicitly, as in \[A = \mbox{the set of natural numbers not exceeding 7} = \{1,2,3,4,5,6,7\}.\] For sets with more elements, show the first few entries to display a pattern, and use an ellipsis to indicate “and so on.” For example, \[\{1,2,3,\ldots,20\}\] represents the set of the first 20 positive integers. The repeating pattern can be extended indefinitely, as in \[\begin{aligned} \mathbb{N} &=& \{1,2,3,\ldots\} \\ \mathbb{Z} &=& \{\ldots,-2,-1,0,1,2,\ldots\} \end{aligned}\]

 

In regards to parity, an integer is either even or odd. For now, we will use our common understanding of even and odd and define these terms later in this text. The set of even integers can be described as \(\{\ldots,-4,-2,0,2,4,\ldots\}\).

Set-Builder Notation

We can use a set-builder notation to describe a set. For example, the set of natural numbers is defined as \[\mathbb{N} = \{x\in\mathbb{Z} \mid x>0 \}.\] Here, the vertical bar \(\mid\) is read as “such that” or “for which.” Hence, the right-hand side of the equation is pronounced as “the set of \(x\) belonging to the set of integers such that \(x>0\),” or simply “the set of integers \(x\) such that \(x>0\).” In general, this descriptive method appears in the format \[\{\,\mbox{membership}\;\mid\;\mbox{properties}\,\}.\] The notation \(\mid\) means “such that” or “for which” only when it is used in the set notation. It may mean something else in a different context. Therefore, do not write “let \(x\) be a real number \(\mid\) \(x^2>3\)” if you want to say “ let \(x\) be a real number such that \(x^2>3\).” It is considered improper to use a mathematical notation as an abbreviation.

Example \(\PageIndex{3}\)

Write these two sets \[\{x\in\mathbb{Z} \mid x^2 \leq 1\} \quad\mbox{and}\quad \{x\in\mathbb{N} \mid x^2 \leq 1\}\] by listing their elements explicitly.

Solution

The first set has three elements, and equals \(\{-1,0,1\}\). The second set is a singleton set; it is equal to \(\{1\}\).

hands-on exercise \(\PageIndex{1}\label{he:setintro-01}\)

Use the roster method to describe the sets \(\{x\in\mathbb{Z} \mid x^2\leq20\}\) and \(\{x\in\mathbb{N} \mid x^2\leq20\}\).

hands-on exercise \(\PageIndex{2}\label{he:setintro-02}\)

Use the roster method to describe the set \[\{x\in\mathbb{N} \mid x\leq20 \mbox{ and $x=n^2$ for some integer $n$}\}.\]

There is a slightly different format for the set-builder notation. Before the vertical bar, we describe the form the elements assume, and after the vertical bar, we indicate from where we are going to pick these elements: \[\{\,\mbox{pattern}\;\mid\;\mbox{membership}\,\}.\] Here the vertical bar \(\mid\) means “where.” For example, \[\{ x^2 \mid x\in\mathbb{Z} \}\] is the set of \(x^2\) where \(x\in\mathbb{Z}\). It represents the set of squares: \(\{0,1,4,9,16,25,\ldots\}\).

Example \(\PageIndex{4}\)

The set \[\{ 2n \mid n\in\mathbb{Z} \}\] describes the set of even numbers. We can also write the set as \(2\mathbb{Z}\).

hands-on exercise \(\PageIndex{3}\label{he:setintro-03}\)

Describe the set \(\{2n+1 \mid n\in\mathbb{Z}\}\) with the roster method.

hands-on exercise \(\PageIndex{4}\label{he:setintro-04}\)

Use the roster method to describe the set \(\{3n \mid n\in\mathbb{Z}\}\).

 

Interval Notation

An interval is a set of real numbers, all of which lie between two real numbers. Should the endpoints be included or excluded depends on whether the interval is open, closed, or half-open. We adopt the following interval notation to describe them: \[\displaylines{ (a,b) = \{x\in\mathbb{R} \mid a < x < b \}, \cr [a,b] = \{x\in\mathbb{R} \mid a\leq x\leq b \}, \cr [a,b) = \{x\in\mathbb{R} \mid a\leq x < b \}, \cr (a,b] = \{x\in\mathbb{R} \mid a < x\leq b \}. \cr}\] It is understood that \(a\) must be less than  \(b\). Hence, the notation \((5,3)\) does not make much sense. How about \([3,3]\)? This may be used in some texts to mean \(\{3\}\) but we will only use \(a < b\) for intervals and use roster notation for  single number such as \(\{3\}\).

An interval contains not just integers, but all real numbers between the two endpoints.  For instance, \((1,5)\mathbb \neq \{2,3,4\}\) because the interval \((1,5)\) also includes real numbers such at \(1.276\), \(\sqrt{2}\), and \(\pi\).

We can use \(\pm\infty\) in the interval notation: \[\begin{aligned} (a,\infty) &=& \{ x\in\mathbb{R} \mid a<x \}, \\ (-\infty,a) &=& \{ x\in\mathbb{R} \mid x<a \}. \end{aligned}\] However, we cannot write \((a,\infty]\) or \([-\infty,a)\), because \(\pm\infty\) are not numbers. It is nonsense to say \(x\leq\infty\) or \(-\infty\leq x\). For the same reason, we can write \([a,\infty)\) and \((-\infty,a]\), but not \([a,\infty]\) or \([-\infty,a]\).

Example \(\PageIndex{5}\)

Write the intervals \((2,3)\), \([2,3]\), and \((2,3]\) in the descriptive form.

Solution

According to the definition of an interval, we find \[\begin{aligned} {(2,3)} &=& \{x\in\mathbb{R} \mid 2<x<3\}, \\ {[2,3]} &=& \{x\in\mathbb{R} \mid 2\leq x\leq 3\}, \\ {(2,3]} &=& \{x\in\mathbb{R} \mid 2 < x\leq 3\}. \end{aligned}\] What would you say about \([2,3)\)?

Example \(\PageIndex{6}\)

Write these sets \[\{x\in\mathbb{R} \mid -2 \leq x < 5\} \quad\mbox{and}\quad \{x\in\mathbb{R} \mid x^2 \leq 1\}\] in the interval form.

Solution

The answers are \([-2,5)\) and \([-1,1]\), respectively. The membership of \(x\) affects the answers. If we change the second set to \(\{x\in\mathbb{Z} \mid x^2\leq 1\}\), the answer would have been \(\{-1,0,1\}\). Can you explain why \(\{-1,0,1\} \mathbb \neq [-1,1]\)?

Example \(\PageIndex{7}\)

Be sure you are using the right types of numbers. Compare these two sets \[\begin{aligned} S &=& \{x\in\mathbb{Z} \mid x^2 \leq 5 \}, \\ T &=& \{x\in\mathbb{R} \mid x^2 \leq 5 \}. \end{aligned}\] One consists of integers only, while the other contains real numbers. Thus, \(S=\{-2,-1,0,1,2\}\), and \(T=\big[-\sqrt{5},\sqrt{5}\,\big]\).

Note

If the membership is not specified, such as: \( \{x \; | \; x^2 \leq 5 \} \) then it is understood that \(\mathbb{R}\) is the default set that \(x\) belongs to.

 

hands-on exercise \(\PageIndex{5}\label{he:setintro-05}\)

Which of the following sets \[\{x\in\mathbb{Z} \mid 1<x<7\} \quad\mbox{and}\quad \{x \mid 1<x<7\}\] can be represented by the interval notation \((1,7)\)? Explain.

hands-on exercise \(\PageIndex{6}\label{he:setintro-06}\)

Explain why \([2,7\,]\mathbb \neq\{2,3,4,5,6,7\}\).

hands-on exercise \(\PageIndex{7}\label{he:setintro-07}\)

True or false: \((-2,3)=\{-1,0,1,2\}\)? Explain.

Let \(S\) be a set of numbers; we define \[\begin{aligned} S^+ &=& \{ x\in S \mid x>0 \}, \\ S^- &=& \{ x\in S \mid x<0 \}, \\ S^* &=& \{ x\in S \mid x\mathbb \neq 0 \}. \end{aligned}\] In plain English, \(S^+\) is the subset of \(S\) containing only those elements that are positive, \(S^-\) contains only the negative elements of \(S\), and \(S^*\) contains only the nonzero elements of \(S\).

Example \(\PageIndex{8}\)

It should be obvious that \(\mathbb{N}=\mathbb{Z}^+\).

hands-on exercise \(\PageIndex{8}\label{he:setintro-08}\)

What is the notation for the set of negative integers?

Some mathematicians also adopt these notations: \[\begin{aligned} bS &=& \{ bx \mid x\in S \}, \\ a+bS &=& \{ a+bx \mid x\in S \}. \end{aligned}\] Accordingly, we can write the set of even integers as \(2\mathbb{Z}\), and the set of odd integers can be represented by \(1+2\mathbb{Z}\).

Example \(\PageIndex{9}\)

\[5\mathbb{Z}=\{\ldots , -15, -10, -5, 0, 5, 10, 15, \ldots\}\]

There are three kinds of real numbers: positive, negative and  zero.

Trichotomy Property

For any two real numbers, \(a\) and \(b\) one and only one of these relations is true:

• \(a <b\)
• \(a=b\)
• \(a>b.\)

Exercises

Exercise \(\PageIndex{1}\)

Determine whether these statements are true or false:

1. \(0\in\mathbb{Q}\)
2. \(0\in\mathbb{Z}\)
3. \(-4\in\mathbb{Z}\)
4. \(-4\in\mathbb{N}\)
5. \(2\in3\mathbb{Z}\)
6. \(-18\in3\mathbb{Z}\)

Answer

(a) true (b) true (c) true (d) false (e) false (f) true

Exercise \(\PageIndex{2}\)

Determine whether these statements are true or false:

1. \(\sqrt{2}\in\mathbb{Z}\)
2. \(-1\notin\mathbb{Z}^+\)
3. \(0\in\mathbb{N}\)
4. \(\pi\in\mathbb{R}\)
5. \(\frac{4}{2}\in\mathbb{Q}\)
6. \(1.5\in\mathbb{Q}\)

Exercise \(\PageIndex{3}\label{ex:prop-03}\)

Explain why \(7\mathbb{Q}=\mathbb{Q}\). Is it still true that \(0\mathbb{Q} = \mathbb{Q}\)?

Answer

By definition, a rational number can be written as a ratio of two integers. After multiplying the numerator by 7, we still have a ratio of two integers. Conversely, given any rational number \(x\), we can multiply the denominator by 7, we obtain another rational number \(y\) such that \(7y=x\). Hence, the two sets \(7\mathbb{Q}\) and \(\mathbb{Q}\) contain the same collection of rational numbers. In contrast, \(0\mathbb{Q}\) contains only one number, namely, 0. Therefore, \(0\mathbb{Q}\neq\mathbb{Q}\).

Exercise \(\PageIndex{4}\label{ex:prop-4}\)

Find the number(s) \(k\) such that \(k\mathbb{Z}=\mathbb{Z}\).

Exercise \(\PageIndex{5}\)

Determine whether these statements are true or false:

(See section on Closure.)

1. The set of natural numbers is closed under subtraction.
2. The set of integers is closed under subtraction.
3. The set of integers is closed under division.
4. The set of rational numbers is closed under subtraction.
5. The set of rational numbers is closed under division.
6. \(\mathbb{Q^*}\) is closed under division.

Answer

(a) false (b) true (c) false (d) true (e) false (f) true