In these lessons, we will learn the intersection of three sets, how to shade regions of Venn Diagrams
involving three sets and how to solve problems using the Venn Diagram of three sets (three circles).

The **intersection** of three sets *X*, *Y* and Z is the set of elements that are common to sets *X*, *Y* and *Z*. It is denoted by *X* ∩ *Y* ∩ *Z*

**Example: **

Draw a Venn diagram to represent the relationship between the sets

* X* = {1, 2, 5, 6, 7, 9}, *Y* = {1, 3, 4, 5, 6, 8} and

* Z * = {3, 5, 6, 7, 8, 10}

**Solution: **

We find that *X* ∩ *Y* ∩ *Z* = {5, 6}, *X * ∩ *Y* = {1, 5, 6},

* Y* ∩ *Z* = {3, 5, 6, 8} and *X* ∩ *Z* = {5, 6, 7}

For the Venn diagram:

**Step 1 **: Draw three overlapping circles to represent the three sets.

** Step 2** : Write down the elements in the intersection *X* ∩ *Y* ∩ *Z*

** Step 3** : Write down the remaining elements in the intersections:

* X* ∩ *Y, Y* ∩ *Z * and *X* ∩ *Z*

** Step 4** : Write down the remaining elements in the respective sets.

Again, notice that you start filling the Venn diagram from the elements in the intersection first.

In general, there are many ways that 3 sets may intersect. Some examples are shown below.

This video shows how to shade regions of Venn Diagrams involving three sets.

Example:

Shade the indicated region:

1) (A ∩ B) ∩ C

2) (A ∪ B) ∩ C Venn Diagrams: Shading Regions with Three Sets, Part 2 of 2

More example to show to shade regions of Venn Diagrams involving three sets.

Example:

Shade the indicated region:

3) (A ∪ B)' ∩ C

4) (A' ∩ B') ∩ C'

Create an expression to represent the outlines part of the Venn Diagram shown.

Give the elements in (A ∪ B') ∩ C

Use the given information to fill in the number of elements in each region of the Venn Diagram.

This video solves two problems using Venn Diagrams. One with two sets and one with three sets.

Example 1:

150 college freshmen were interviewed.

85 were registered for a math class

70 were registered for an English class

50 were registered for both math and English

1. How many signed up only for a math class?

1. How many signed up only for an English class?

1. How many signed up for math or English?

1. How many signed up for neither math nor English?

Example 2:

100 were students interviewed

28 took PE

31 took Bio

42 took Eng

9 took PE and Bio

10 took PE and Eng

6 took Bio and Eng

4 took all three subjects

How many students took none of the three subjects?

How many students took PE, but not Bio or Eng?

How many students took Gio and PE but not Eng?

Example:

110 college freshmen were surveyed

25 took physics

45 took biology

45 took mathematics

10 took physics and mathematics

8 took biology and mathematics

6 took physics and biology

5 took all three

a. How many students took biology, but neither physics nor mathematics?

b. How many students took biology, physics or mathematics?

c. How many students did not take any of the three subjects?

In this video we go over a basic word problem involving three sets. We use a Venn diagram to answer the series of questions.

Enter an expression like (A Union B) Intersect (Complement C) to describe a combination of two or three sets and get the notation and Venn diagram. Use parentheses, Union, Intersection, and Complement

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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