Learning Objectives
The Basics of Sets
- Elements
- Empty set
- Cardinality
- Forms of writing sets
- Description form
- Roster form
- Set Builder Notation form
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.
recall sets of real numbers
Recall the sets of real numbers you studied previously. Each number contained in a set is an element of the set that contains it. For example, the number [latex]1[/latex] is an element of the set of counting numbers. The number [latex]\dfrac{2}{3}[/latex] is an element of the set of rational numbers. And so on. The same idea applies to any set of distinct objects, as described below.
Set
A set is a collection of distinct objects, called elements of the set
A set can be defined by describing the contents, or by listing the elements of the set, enclosed in braces.
A set is well defined if its elements are clearly described. There’s no question about what the set’s elements are.
Example
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 listing the elements of the set:
- {1, 3, 9, 12}
- {red, orange, yellow, green, blue, indigo, purple}
Notation
Commonly, we will use a capital letter to represent a set, to make it easier to refer to that set later.
Element
The objects used to form a set are called its element or its members.
Let A = {v, w, x, y, z}
Here ‘A’ is the name of the set whose elements (members) are v, w, x, y, z.
Notation
The symbol ∈ means “is an element of” and the symbol ∉ means “is not an element of”
Examples
If a set A = {3, 6, 9, 10, 13, 18}. State whether the following statements are ‘true’ or ‘false’:
a) 7 ∈ A
b) 10 ∉ A
c) 13 ∈ A
d) 9, 10 ∈ A
SOLUTIONS:
a) False, since the element 7 does not belong to the given set A.
b) False, since the element 10 belongs to the given set A.
c) True, since the element 13 belongs to the given set A.
d) True, since the elements 9 and 10 both belong to the given set A.
EMPTy Set
A set that does not contain any elements is called an empty set or null set.
Notation
An empty set is represented as { }, containing no element at all. It is also represented using the symbol ∅ (read as ‘phi’).
Examples
Let’s consider the following examples where we need to determine if the given sets are empty sets.
a) A month with 33 days
Since there are no months with 33 days, we can conclude that this is an empty set.
b) X = {x | x is a prime number and 14<x<16}
Let’s start by making a list of prime numbers 2, 3, 5, 7, 11, 13, 17…,
We can see that there are no prime numbers between 14 and 16. We can conclude that this is an empty set.
Cardinality
The cardinality of a set is basically the size of the set. Cardinality is simply the number of elements in the set.
Notation
n(A) means the number of members in set A.
ExAMPLES
Find the cardinality of the following sets
a) Let A = {1, 2, 3, 4, 5, 6}, n(A) = 6
b) Let S be the letters in the word bubble. n(S) = 4. The answer is 4 because the set is S = {b, u, l, e} which has 4 elements in it.
c) Let B = ∅. n(B) = 0. The answer is 0 because the set is empty.
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}.
Sets can be listed in three ways: Description form, Roster form and Set Builder Notation form.
Set builder what? Yeah, we’ll get there in a second. First, let’s go over Roster form. Have you ever played on a
team? The list of people on the team is the Team Roster.
Roster form is just a way of listing the elements.
For example: If the Set A is “the set of Natural numbers less than 6,” it would be written in Roster form as:
A={1, 2, 3, 4, 5}
Description form is simply describing the elements in the set, like, Set A is the “Set of days that end in Y.”
A={ Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}
And last but not least, the dreaded Set Builder Notation form. This is probably the trickiest of the three ways to notate a set, but it’s actually pretty fun once you get the hang of it. It’s kind of like cracking a code. It’s a statement and each part of the statement is telling you something.
Example:
Set A is the set of all x such that x is an element of the Natural numbers and x is less than 100. What the…? Looks like this: A={x|x ∈ N and x<100} The answer would be A= { 2, 3, 4, 5,…, 99}
Break it down like this:
A “A is the set”
{x “Of all x”
|x “Such that x”
∈ N “Is an element of the Natural numbers”
And x < “And x is less than 100”
See, that wasn’t so hard.
Here’s another example: Write the following in set builder notation. X is a letter in the word MATH.
We’d want to start with braces, then the variable then a vertical line.
{x|
Next, we want to say what x is an element of. In this case it is a letter.
{x|x∈English Letters, all letters in MATH}
This is the end of the section. Close this tab and proceed to the corresponding assignment.
