3.1: Basics of Sets
- Page ID
- 74301
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)3.1 Learning Objectives
- Use the correct notation for elements of sets and subsets
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Find the cardinality of a set, including the empty set
An art collector might own a collection of paintings, while a music lover might keep a collection of CDs. Any collection of items can form a set.
Set
Definition: Set
A set is a collection of distinct objects, called elements of the set
A set is said to be well-defined if the contents of the set can be easily determined by the description.
Definition: Roster Method and Set-Builder
A set can be defined using word descriptions, roster method, and set-builder notation.
A word description could be: The days of the week.
The roster method lists the elements of the set enclosed in curly brackets.
\(\{Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday\}\)
Set-builder notation describes what a general element should be.
\(\{x|x=\text{a day of the week}\}\). In English this reads as: "The set of all x such that x is a day of the week."
Example 1
Some examples of sets defined by describing the contents:
- The set of all even numbers
- The set of all books written about travel to Chile
Some examples of sets defined by roster method:
- {1, 3, 9, 12}
- {red, orange, yellow, green, blue, indigo, purple}
Some examples of sets defined by set-builder notation:
- \(\{x| 0\leq x\leq 3 \text{ and } x\in \text{integers}\}\)
- \(\{m| m=\text{ a car model}\}\)
A set simply specifies the contents; order is not important. The set represented by {1, 2, 3} is equivalent to the set {3, 1, 2}.
Notation
Commonly, we will use a capital letter to represent a set to make it easier to refer to that set later.
The symbol \(\in\) means “is an element of”.
The symbol \(\notin\) means “is not an element of”.
A set that contains no elements, \(\{ \}\), is called the empty set and is notated Ø or \(\{ \}\).
Example 2
Let \(A=\{1,2,3,4\}\)
To notate that 2 is an element of the set, we would write \(2 \in A\)
To notate that 5 is not an element of the set, we would write \(5 \notin A\)
Sometimes a collection might not contain all the elements of a set. For example, Chris owns three Madonna albums. While Chris’s collection is a set, we can also say it is a subset of the larger set of all Madonna albums.
Definition: Subset
A subset of a set \(A\) is another set that contains only elements from the set \(A\) but may not contain all the elements of \(A\).
If \(B\) is a subset of \(A,\) we write \(B \subseteq A\).
A proper subset is a subset that is not identical to the original set. There is at least one element in the original set that is not in the subset.
If \(B\) is a proper subset of \(A\), we write \(B \subset A\).
To indicate a set \(B\) is not a subset of \(A\), we write \(B \nsubseteq A\)
The empty set is a subset of every set.
Example 3
Consider these three sets
\(A=\) the set of all even numbers\(\quad B=\{2,4,6\} \quad C=\{2,3,4,6\}\)
Here \(B \subset A\) since every element of \(B\) is also an even number, so it is an element of \(A\).
More formally, we could say \(B \subset A\) since if \(x \in B,\) then \(x \in A\)
It is also true that \(B \subset C\).
\(C\) is not a subset of \(A\), since \(C\) contains an element, 3 , that is not contained in \(A\)
Example 4
Suppose a set contains the plays “Much Ado About Nothing”, “MacBeth”, and “A Midsummer’s Night Dream”. What is a larger set this might be a subset of?
Solution
There are many possible answers here. One would be the set of plays by Shakespeare. This is also a subset of the set of all plays ever written. It is also a subset of all British literature.
Try it Now 1
The set \(A=\{1,3,5\} .\) What is a larger set this might be a subset of?
- Answer
-
There are several answers: The set of all odd numbers less than 10. The set of all odd numbers. The set of all integers. The set of all real numbers.
Example 5
If A = {1, 2, 3} , B = {1,3} , and C = {1,2,3}, determine if:
a. \(B \subset A\)
b. \(C \subset A\)
Solution
a. Since B is not an identical set to A, we say that this is a proper subset of A.
b. Since C is an identical set to A, we say that C is not a proper subset of A, and should be written as \(C \subseteq A\).
In some situations, we are interested in how many elements a set has. This number is called the cardinality of the set.
Definition: Cardinality
The cardinality of a set is the number of elements in the set.
n(A) is the notation used for the cardinality of the set A.
Example 6
Given A = { 1, 3, 5 }, find the cardinality for the set A.
Solution
Since there are 3 elements in the set A, then we say n(A) = 3.
Example 7
Given the set is empty, find the cardinality for the empty set.
Solution
Since there are no elements in the empty set { }, then we say n(Ø) = 0.
We say that two sets are equal only when they contain exactly the same elements. If they have the same cardinality but not the exact same elements, the sets are called equivalent.
Example 8
Determine if the sets are equal, equivalent, or neither.
- {1, 3, 9, 12} and {12, 3, 1, 9}
- {red, orange, yellow, green, blue, indigo, purple} and {blue, indigo, purple}
- {a, e, i, o, u} and {p, q, r, s, t}
Solution
- These sets are equal since it has the same elements even though they are in a different order.
- This set is neither equal or equivalent.
- This set is equivalent since it contains the same number of elements but they are different.