### Learning Outcomes

- Describe memberships of sets, including the empty set, using proper notation, and decide whether given items are members and determine the cardinality of a given set.
- Describe the relations between sets regarding membership, equality, subset, and proper subset, using proper notation.
- Perform the operations of union, intersection, complement, and difference on sets using proper notation.
- Be able to draw and interpret Venn diagrams of set relations and operations and use Venn diagrams to solve problems.
- Recognize when set theory is applicable to real-life situations, solve real-life problems, and communicate real-life problems and solutions to others.

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

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

### 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

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

To notate that 2 is element of the set, we’d write 2 ∈ *A*

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}.

## Subsets

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 excludes at least one element of the original set.

If *B* is a proper subset of *A*, we write *B* ⊂ *A*

### Example

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*

### Try It

### Example

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?

### Try It

Consider the set [latex]A = \{1, 3, 5\} [/latex]. Which of the following sets is [latex]A [/latex] a subset of?

[latex]X = \{1, 3, 7, 5\} [/latex]

[latex]Y = \{1, 3 \} [/latex]

[latex]Z = \{1, m, n, 3, 5\}[/latex]

### Exercises

Given the set: *A* = {*a*, *b*, *c*, *d*}. List all of the subsets of *A
*

Listing the sets is fine if you have only a few elements. However, if we were to list all of the subsets of a set containing many elements, it would be quite tedious. Instead, in the next example we will consider each element of the set separately.

### Example

In the previous example, there are four elements. For the first element, *a*, either it’s in the set or it’s not. Thus there are 2 choices for that first element. Similarly, there are two choices for *b*—either it’s in the set or it’s not. Using just those two elements, list all the possible subsets of the set {a,b}

Now let’s include *c, *just for fun. List all the possible subsets of the new set {a,b,c}.

Again, either *c* is included or it isn’t, which gives us two choices. The outcomes are {}, {*a*}, {*b*}, {*c*}, {*a*,*b*}, {*a*,*c*}, {*b*,*c*}, {*a*,*b*,*c*}. Note that there are [latex]2^{3}=8[/latex] subsets.

If you include four elements, there would be [latex]2^{4}=16[/latex] subsets. 15 of those subsets are proper, 1 subset, namely {*a*,*b*,*c*,*d*}, is not.

In general, if you have *n* elements in your set, then there are [latex]2^{n}[/latex] subsets and [latex]2^{n}−1[/latex] proper subsets.