Lesson Objectives

- Learn the definition of a set in math
- Learn how to list the elements of a set using the roster method
- Learn how to notate a null or empty set using: "{}" or "∅"
- Learn how to use: "∈" to denote a member of a set
- Learn how to use: "∉" to denote is not a member of a set
- Learn about the universal set
- Learn about the difference between finite and infinite sets
- Learn about proper and improper subsets
- Learn how to determine how many subsets can be made from a given set

## What is a Set in Math?

Over the course of our math studies, we will often come across the topic of sets. When used in Algebra, a set is simply a collection of things. Most often, we will see a solution set. A solution set is a set which contains all the solutions to a given equation. Suppose we had the following equation:

4x - 1 = 11

This equation has a solution of:

x = 3

Therefore, our solution set would contain one element or member, the number 3. We can show our solution set as:

{3}

As an example, we could have a set which contains all states in the U.S. which start with the letter "A":

{Alabama, Alaska, Arizona, Arkansas}

We can see this set contains four members or elements. These are listed using the roster method, meaning the elements are separated by a comma and are placed between set braces.

Example 1: List the elements of each set using the roster method

The set of even whole numbers between 1 and 9:

{2, 4, 6, 8}

When we list the elements of a set, the order is not important. If we have a set which contains the elements: 3 and 9, it can be listed as:

{3, 9} or {9, 3}

We can use a capital letter to name a set. Let set C be the set whose elements are the whole numbers that are less than 7:

C = {0, 1, 2, 3, 4, 5, 6}

{} "empty set braces"

∅ "empty set symbol"

What we don't want to do is combine the two:

{∅} "THIS IS WRONG!!!!"

Example 2: List the elements of the set using the roster method

Let set A be the set of turtles living on the planet Mars:

A = {} or A = ∅

Since there are no turtles on the planet Mars (at least none as of right now or that we know about), this set contains no elements. Set A is, therefore, an empty or null set.

Z = {1, 3, 5, 7, 9}

We can show that 3 is a member of set Z using the following notation:

3 ∈ Z (3 is a member of set Z)

Z = {1, 3, 5, 7, 9}

We can show that 4 is not a member of set Z using the following notation:

4 ∉ Z (4 is not a member of set Z)

Example 3: Determine if each statement is true or false

A = {3, 12, 16, 19}

2 ∈ A

This is false since 2 is not an element of set A

Example 4: Determine if each statement is true or false

B = {-9, -1, 18, 22}

9 ∉ B

This is true since 9 is not an element of set B

M = {California, Arizona, New Mexico, Texas}

Our set M contains very specific features, each element is a state within the U.S. that borders Mexico. If we think about the universal set here, it will be all of the states in the U.S.:

U = {Alabama, Alaska, ... , Wisconsin, Wyoming}

W = {0, 1, 2, 3, 4, ...}

The three dots "..." is known as an ellipsis. The ellipsis indicates that our pattern continues forever. This means after 4 comes 5, then 6, and so on and so forth. Essentially, there is no largest whole number. We can keep adding 1 to the previous whole number to obtain the next.

We will also come across finite sets. A finite set has a countable number of elements. Suppose we say that set P is the set of whole numbers less than 10:

P = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

P is an example of a finite set since there are exactly ten elements.

A = {1, 3, 7}

B = {1, 2, 3, 4, 5, 6, 7}

Set A is a subset of set B since set B contains all elements of set A. More specifically, we can say that A is a proper subset of B since the two sets are also not equal to each other.

Proper Subset Symbol: "⊂"

A ⊂ B

When two sets are equal to each other, we can say the two sets are equal or we can say they are improper subsets of each other. Suppose we have the following two sets:

Set D = {1, 9 , 13}

Set N = {9, 13, 1}

Since these two sets contain exactly the same elements, they are considered "equal sets" or "improper subsets":

Improper Subset Symbol: "⊆"

D = N or D ⊆ N

Additionally, we have a symbol to show that a given set is not a subset of another:

Not a Subset: "⊄"

As an example, suppose we have two sets:

L = {1, 8, 10}

Z = {2, 8, 10, 12}

Set L is not a subset of set Z since Z does not contain all elements of L (missing a 1):

L ⊄ Z

Example 5: Determine if each statement is true or false

A = {1, 2, 3, 4, 5}

B = {1, 3, 5}

D = {2, 5, 6}

B ⊂ A

This is true, set B is a proper subset of set A. Set A contains all elements of set B (1, 3, 5) and the two sets are not equal.

D ⊄ A

This is true, set D is not a proper subset of set A. Set A does not contain the number 6.

B ⊂ D

This is false, set B is not a proper subset of set D.

D = {turkey, rice, peas}

How many subsets (either proper or improper) can be made from set D?

Our general rule:

2

Our set D has three items, which means there are 8 (2

{turkey},{rice},{peas}

{turkey, rice},{turkey, peas},{rice, peas}

{turkey, rice, peas}

∅

The last subset, which is the empty set is quite confusing. Why is the empty set a subset of set D? The empty set is defined as a subset of any set. To make this easy to understand, think about each subset as a possible choice. We could choose to eat all of the items on the dinner menu, only two items, or only one item. Lastly, we could choose to eat nothing. Since this is a possibility, the empty set is a subset.

Example 6: Determine how many subsets can be made from the given set

Q = {blue, red, green, yellow}

Since set Q has four elements, we can plug in a 4 for n in our formula:

2

Set Q has 16 possible subsets:

{blue},{red},{green},{yellow}

{blue, red},{blue, green},{blue, yellow}

{red, green},{red, yellow}

{green, yellow}

{blue, red, green}

{blue, red, yellow}

{blue, green, yellow}

{red, green, yellow}

{blue, red, green, yellow}

∅

4x - 1 = 11

This equation has a solution of:

x = 3

Therefore, our solution set would contain one element or member, the number 3. We can show our solution set as:

{3}

### Listing the Elements of a Set Using the Roster Method

As we alluded to above, the items in a set are known as members or elements of the set. In many cases, we will just list the elements of the set inside of set braces (curly brackets "{}"). When we list elements of a set in this manner, it is referred to as the "roster method".As an example, we could have a set which contains all states in the U.S. which start with the letter "A":

{Alabama, Alaska, Arizona, Arkansas}

We can see this set contains four members or elements. These are listed using the roster method, meaning the elements are separated by a comma and are placed between set braces.

Example 1: List the elements of each set using the roster method

The set of even whole numbers between 1 and 9:

{2, 4, 6, 8}

When we list the elements of a set, the order is not important. If we have a set which contains the elements: 3 and 9, it can be listed as:

{3, 9} or {9, 3}

We can use a capital letter to name a set. Let set C be the set whose elements are the whole numbers that are less than 7:

C = {0, 1, 2, 3, 4, 5, 6}

### Empty or Null Set

In some cases, we will have a set that contains no elements. This type of set is referred to as an empty or null set. When we have an empty or null set, we can use either type of notation:{} "empty set braces"

∅ "empty set symbol"

What we don't want to do is combine the two:

{∅} "THIS IS WRONG!!!!"

Example 2: List the elements of the set using the roster method

Let set A be the set of turtles living on the planet Mars:

A = {} or A = ∅

Since there are no turtles on the planet Mars (at least none as of right now or that we know about), this set contains no elements. Set A is, therefore, an empty or null set.

### Using "∈" to Denote a Member of a Set

The symbol "∈" is used to denote "is a member of the set". As an example, suppose we have set Z:Z = {1, 3, 5, 7, 9}

We can show that 3 is a member of set Z using the following notation:

3 ∈ Z (3 is a member of set Z)

### Using "∉" to Denote is not a Member of a Set

The symbol "∉" is used to denote "is not a member of the set". As an example, suppose we again have set Z:Z = {1, 3, 5, 7, 9}

We can show that 4 is not a member of set Z using the following notation:

4 ∉ Z (4 is not a member of set Z)

Example 3: Determine if each statement is true or false

A = {3, 12, 16, 19}

2 ∈ A

This is false since 2 is not an element of set A

Example 4: Determine if each statement is true or false

B = {-9, -1, 18, 22}

9 ∉ B

This is true since 9 is not an element of set B

### Universal Set

The universal set, normally denoted as "U", is a set that contains all elements under consideration. As an example, suppose we let set M be the states in the U.S. that border Mexico:M = {California, Arizona, New Mexico, Texas}

Our set M contains very specific features, each element is a state within the U.S. that borders Mexico. If we think about the universal set here, it will be all of the states in the U.S.:

U = {Alabama, Alaska, ... , Wisconsin, Wyoming}

### Finite Sets vs Infinite Sets

When we deal with sets in math, many sets will be infinite, meaning there is an unlimited number of elements. As an example, consider the set of whole numbers, which we will represent with the letter W:W = {0, 1, 2, 3, 4, ...}

The three dots "..." is known as an ellipsis. The ellipsis indicates that our pattern continues forever. This means after 4 comes 5, then 6, and so on and so forth. Essentially, there is no largest whole number. We can keep adding 1 to the previous whole number to obtain the next.

We will also come across finite sets. A finite set has a countable number of elements. Suppose we say that set P is the set of whole numbers less than 10:

P = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

P is an example of a finite set since there are exactly ten elements.

### Subsets

In order to explain the concept of a subset, let's take a look at two sets:A = {1, 3, 7}

B = {1, 2, 3, 4, 5, 6, 7}

Set A is a subset of set B since set B contains all elements of set A. More specifically, we can say that A is a proper subset of B since the two sets are also not equal to each other.

Proper Subset Symbol: "⊂"

A ⊂ B

When two sets are equal to each other, we can say the two sets are equal or we can say they are improper subsets of each other. Suppose we have the following two sets:

Set D = {1, 9 , 13}

Set N = {9, 13, 1}

Since these two sets contain exactly the same elements, they are considered "equal sets" or "improper subsets":

Improper Subset Symbol: "⊆"

D = N or D ⊆ N

Additionally, we have a symbol to show that a given set is not a subset of another:

Not a Subset: "⊄"

As an example, suppose we have two sets:

L = {1, 8, 10}

Z = {2, 8, 10, 12}

Set L is not a subset of set Z since Z does not contain all elements of L (missing a 1):

L ⊄ Z

Example 5: Determine if each statement is true or false

A = {1, 2, 3, 4, 5}

B = {1, 3, 5}

D = {2, 5, 6}

B ⊂ A

This is true, set B is a proper subset of set A. Set A contains all elements of set B (1, 3, 5) and the two sets are not equal.

D ⊄ A

This is true, set D is not a proper subset of set A. Set A does not contain the number 6.

B ⊂ D

This is false, set B is not a proper subset of set D.

### Number of Subsets of a Set with n Elements

In some cases, we will want to know how many subsets can be made from a given set. Let's suppose we had a set D, which contains our dinner items:D = {turkey, rice, peas}

How many subsets (either proper or improper) can be made from set D?

Our general rule:

2

^{n}» n = number of elements in the setOur set D has three items, which means there are 8 (2

^{3}= 8) possible subsets. We can list them:{turkey},{rice},{peas}

{turkey, rice},{turkey, peas},{rice, peas}

{turkey, rice, peas}

∅

The last subset, which is the empty set is quite confusing. Why is the empty set a subset of set D? The empty set is defined as a subset of any set. To make this easy to understand, think about each subset as a possible choice. We could choose to eat all of the items on the dinner menu, only two items, or only one item. Lastly, we could choose to eat nothing. Since this is a possibility, the empty set is a subset.

Example 6: Determine how many subsets can be made from the given set

Q = {blue, red, green, yellow}

Since set Q has four elements, we can plug in a 4 for n in our formula:

2

^{n}» 2^{4}= 16Set Q has 16 possible subsets:

{blue},{red},{green},{yellow}

{blue, red},{blue, green},{blue, yellow}

{red, green},{red, yellow}

{green, yellow}

{blue, red, green}

{blue, red, yellow}

{blue, green, yellow}

{red, green, yellow}

{blue, red, green, yellow}

∅

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