Lesson Objectives

- Demonstrate a basic understanding of sets
- Learn how to build a Venn Diagram
- Learn how to find the complement of a set
- Learn how to find the intersection of two or more sets
- Learn how to find the union of two or more sets
- Learn how to identify disjoint sets

## Sets - How to Draw a Venn Diagram

In our last lesson, we learned the basics of sets. In this lesson, we will expand on our current knowledge of sets and learn some additional information. Let's begin by showing how to draw a Venn Diagram. The Venn Diagram is a way to visually represent the relationship between two or more sets. Recall that U is generally used to represent the universal set. This is the set that contains all elements under consideration. For example, if we had a set P, which contained all U.S. Presidents born in the state of California, it would contain one element:

P = {Richard Nixon}

In this case, since we are discussing U.S. Presidents, the universal set U, would contain all U.S. Presidents. As of 2019:

U = {George Washington, John Adams, ... Barack Obama, Donald Trump}

Of course, set P and set U would change over time. To draw a Venn Diagram, we start by drawing a rectangle. The rectangle represents the universal set. All other sets are drawn as regions within the rectangle. Let's look at an example.

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

A = {2, 3, 4}

B = {4, 5, 6} From the Venn Diagram, we can immediately see that both set A and set B are subsets of U.

A ⊂ U (A is a subset of set U)

B ⊂ U (B is a subset of set U)

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

A = {2, 3, 4}

B = {4, 5, 6}

The complement of set A, which is denoted as:

A' or A

contains all elements of U that are not in set A.

A' = {1, 5, 6, 7}

The elements 1, 5, 6, and 7 are not in set A but are in set U. These are the elements of A' or the complement of set A.

B' = {1, 2, 3, 7}

The elements 1, 2, 3, and 7 are not in set B but are in set U. These are the elements of B' or the complement of set B.

Visually, if we want A', we can look at our Venn diagram in the region outside of set A: To make it easier to see, we have shown set A with a white background and everything outside of set A with a yellow background. It is easy to see, that the elements 1, 5, 6, and 7, belong to the set that is the complement of set A.

Intersection Symbol: "∩" (placed between two set names or two sets)

If we stay with our above example:

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

A = {2, 3, 4}

B = {4, 5, 6}

A ∩ B = {4}

The intersection of set A and set B is a set that contains one element, 4. This is the only element that is in both sets.

We can easily find the intersection of two sets using a Venn Diagram: The intersection of the two sets A and B is shown on the Venn Diagram as the area shaded yellow.

Union Symbol: "∪" (placed between two set names or two sets)

Again using our above example:

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

A = {2, 3, 4}

B = {4, 5, 6}

A ∪ B = {2, 3, 4, 5, 6}

To build this set, we list all elements of set A and then list all elements of set B that are not already listed. Notice how 4, which is an element that is a member of both sets is only listed once. We can easily find the union of two sets using a Venn Diagram: The union of the two sets A and B is shown on the Venn Diagram as the area shaded yellow.

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

A = {1, 2, 3}

B = {4, 5, 6}

A ∩ B = ∅

In this scenario, A and B have no elements in common. They are disjoint sets. We can easily show this with a Venn Diagram: We can see from our Venn Diagram that there is no overlap between the two sets A and B. This visually tells us they are "disjoint" sets.

Example 1: Determine if each statement is true or false 1) Steven ∈ A

True, Steven is an element of set A

2) B ⊂ C

False, Set B is not a subset of set C

3) A' = {Carlee, Aya, Jim}

False, A' contains the elements: Jennifer, Beth, Carlee, Aya, and Jim

4) A ∩ B = ∅

True, A and B are disjoint sets.

5) A ∪ C = {Steven, Bob, Aya, Jim}

True, Set A contains the elements: Steven and Bob, while set C contains the elements: Aya and Jim

6) U = {Jennifer, Beth}

False, the universal set contains all elements under consideration (all elements in the rectangle).

U = {Jennifer, Beth, Steven, Bob, Carlee, Aya, Jim}

7) C' = {Jennifer, Beth, Steven, Bob, Carlee}

True, set C does not contain the elements: Jennifer, Beth, Steven, Bob, and Carlee.

8) B ∪ C = {Aya}

False, the union of sets B and C is a set that contains three elements: Carlee, Aya, and Jim

P = {Richard Nixon}

In this case, since we are discussing U.S. Presidents, the universal set U, would contain all U.S. Presidents. As of 2019:

U = {George Washington, John Adams, ... Barack Obama, Donald Trump}

Of course, set P and set U would change over time. To draw a Venn Diagram, we start by drawing a rectangle. The rectangle represents the universal set. All other sets are drawn as regions within the rectangle. Let's look at an example.

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

A = {2, 3, 4}

B = {4, 5, 6} From the Venn Diagram, we can immediately see that both set A and set B are subsets of U.

A ⊂ U (A is a subset of set U)

B ⊂ U (B is a subset of set U)

### Complement of a Set

Let's suppose we continue with our same example from above:U = {1, 2, 3, 4, 5, 6, 7}

A = {2, 3, 4}

B = {4, 5, 6}

The complement of set A, which is denoted as:

A' or A

^{c}contains all elements of U that are not in set A.

A' = {1, 5, 6, 7}

The elements 1, 5, 6, and 7 are not in set A but are in set U. These are the elements of A' or the complement of set A.

B' = {1, 2, 3, 7}

The elements 1, 2, 3, and 7 are not in set B but are in set U. These are the elements of B' or the complement of set B.

Visually, if we want A', we can look at our Venn diagram in the region outside of set A: To make it easier to see, we have shown set A with a white background and everything outside of set A with a yellow background. It is easy to see, that the elements 1, 5, 6, and 7, belong to the set that is the complement of set A.

### Intersection of Two Sets

The intersection of two sets is a set that contains all elements that are common to both sets.Intersection Symbol: "∩" (placed between two set names or two sets)

If we stay with our above example:

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

A = {2, 3, 4}

B = {4, 5, 6}

A ∩ B = {4}

The intersection of set A and set B is a set that contains one element, 4. This is the only element that is in both sets.

We can easily find the intersection of two sets using a Venn Diagram: The intersection of the two sets A and B is shown on the Venn Diagram as the area shaded yellow.

### Union of Two Sets

The union of two sets is a set which contains all elements of both sets. If the two sets share elements, each shared element is only listed once.Union Symbol: "∪" (placed between two set names or two sets)

Again using our above example:

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

A = {2, 3, 4}

B = {4, 5, 6}

A ∪ B = {2, 3, 4, 5, 6}

To build this set, we list all elements of set A and then list all elements of set B that are not already listed. Notice how 4, which is an element that is a member of both sets is only listed once. We can easily find the union of two sets using a Venn Diagram: The union of the two sets A and B is shown on the Venn Diagram as the area shaded yellow.

### Disjoint Sets

When two sets have no elements in common, the sets are known as "disjoint" sets. Let's modify our example:U = {1, 2, 3, 4, 5, 6, 7}

A = {1, 2, 3}

B = {4, 5, 6}

A ∩ B = ∅

In this scenario, A and B have no elements in common. They are disjoint sets. We can easily show this with a Venn Diagram: We can see from our Venn Diagram that there is no overlap between the two sets A and B. This visually tells us they are "disjoint" sets.

Example 1: Determine if each statement is true or false 1) Steven ∈ A

True, Steven is an element of set A

2) B ⊂ C

False, Set B is not a subset of set C

3) A' = {Carlee, Aya, Jim}

False, A' contains the elements: Jennifer, Beth, Carlee, Aya, and Jim

4) A ∩ B = ∅

True, A and B are disjoint sets.

5) A ∪ C = {Steven, Bob, Aya, Jim}

True, Set A contains the elements: Steven and Bob, while set C contains the elements: Aya and Jim

6) U = {Jennifer, Beth}

False, the universal set contains all elements under consideration (all elements in the rectangle).

U = {Jennifer, Beth, Steven, Bob, Carlee, Aya, Jim}

7) C' = {Jennifer, Beth, Steven, Bob, Carlee}

True, set C does not contain the elements: Jennifer, Beth, Steven, Bob, and Carlee.

8) B ∪ C = {Aya}

False, the union of sets B and C is a set that contains three elements: Carlee, Aya, and Jim

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