Basics

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:

a) The set of all even numbers

b) The set of all books written about travel to Chile

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

a) {1, 3, 9, 12}

b) {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 ∊ means “is an element of”.

A set that contains no elements, { }, is called the empty set and is notated ∅

Example 2

Let A = {1, 2, 3, 4}

To notate that 2 is element of the set, we’d write 2 ∊ 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 BA

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 BA

Example 3

Consider these three sets

A = the set of all even numbers           B = {2, 4, 6}   C = {2, 3, 4, 6}

Here BA since every element of B is also an even number, so is an element of A.

More formally, we could say BA since if x B, then x A.

It is also true that BC.

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?

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?