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 *B* ⊆ *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* ⊂ *A*

### Example 3

Consider these three sets

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

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

More formally, we could say *B* ⊂ *A* since if *x *∊ *B*, then *x *∊ *A*.

It is also true that *B* ⊂ *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?

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?