9.1: Basics of Sets

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

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 1

Some examples of sets defined by describing the contents:

1. The set of all even numbers
2. The set of all books written about travel to Chile

Some examples of sets defined by listing the elements of the set:

1. {1, 3, 9, 12}
2. {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}.

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 \(\emptyset\)

Example 2

Let \(A=\{1,2,3,4\}\)

To notate that 2 is element of the set, we write \(2 \in A\)

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.

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 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 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.

Often times we are interested in the number of items in a set or subset. This is called the cardinality of the set.

Cardinality

The number of elements in a set is the cardinality of that set.

The cardinality of the set \(A\) is often notated as \(|A|\) or \(n(A)\)

Example 5

Let \(A=\{1,2,3,4,5,6\}\) and \(B=\{2,4,6,8\}\)

What is the cardinality of the set \(B\) ? The set \(A\)?

Solution

The cardinality of \(B\) is \(4\), since there are 4 elements in the set.

The cardinality of \(A\) is \(6\), since there are 6 elements in the set.

Example 6

What is the cardinality of the set of names for the months of the year?

Solution

The cardinality of this set is \(12\), since there are 12 months, and thus 12 names of months, in the year.