7.1: 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'd 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.