Section 2.1 Basics
Objectives
Define sets, subsets, and elements.
Discuss the notation used for sets, subsets, and elements.
Compute cardinal numbers of sets.
It is natural for us to classify items into groups, or sets, and consider how those sets overlap with each other. We can use these sets understand relationships between groups, and to analyze survey data.
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.
Definition 2.1.1. 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 curly brackets.
Example 2.1.2.
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}
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}.
Remark 2.1.3. Notation.
Commonly, we will use a variable to represent a set, to make it easier to refer to that set later.
The symbol \(\in\) means “is an element of”.
A set that contains no elements, { }, is called the empty set and is notated \(\emptyset\text{.}\)
Example 2.1.4.
Let A = {1, 2, 3, 4}
To notate that 2 is element of the set, we’d write \(2 \in A\text{.}\)
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 2.1.5. 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 - it contains fewer elements.
If B is a proper subset of A, we write \(B \subset A\)
Example 2.1.6.
Consider these three sets
\(A = \)the set of all even numbers, \(B = \{2, 4, 6\}\text{,}\) \(C = \{2, 3, 4, 6\}\)
Here \(B \subset A\) since every element of \(B\) is also an even number, so is an element of \(A\)
More formally, we could say \(B \subset A\) since if \(x \in B\text{,}\) then \(x \in A\text{.}\)
It is also true that \(B \subset C\text{.}\)
C is not a subset of \(A\text{,}\) since \(C\) contains an element, 3, that is not contained in \(A\text{.}\)
Example 2.1.7.
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.Problem 2.1.8. Try It Now.
The set \(A = \{1, 3, 5\}\text{.}\) What is a larger set this might be a subset of?
Answer.Often times we are interested in the number of items in a set or subset. This is called the cardinality of the set.
Definition 2.1.9. Cardinality.
The number of elements in a set is the cardinality of that set.
The cardinality, or cardinal number, of the set A is often notated as \(n\left(A\right)\text{.}\)
Example 2.1.10.
Find the cardinal number of each set.
\(\displaystyle A = \{1, 2, 3, 5\}\)
\(\displaystyle B = \emptyset\)
There are four elements in set A. The cardinal number of set A is \(n\left(A\right) = 4\)
Since set B is the empty set, it does not contain any elements. \(n\left(B\right)=0\)
Example 2.1.11.
Find the cardinality of the set.
\(A = \{\{1, 2\},3, 5\}\)
Solution.There are three elements, or objects, in set A. Notice that we count \(\{1, 2\}\) as one element.
\(n\left(A\right)=3\)