Sets Intersection: Intersection Of Three Sets

Venn Diagrams Of Three Sets

A Venn diagram uses overlapping circles to represent sets. For three sets, you’ll have three overlapping circles, usually within a rectangle that represents the universal set (all possible elements).

The intersection of the three sets is the region where all three circles overlap. This central, shared area represents the elements that belong to A, B, and C.

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.
Notice that you start filling the Venn diagram from the elements in the intersection first.

Here are the general steps:

1. Draw the Venn Diagram:
   Draw a rectangle to represent the universal set (U).
   Inside the rectangle, draw three overlapping circles, one for each set (A, B, and C).
   
2. Start with the Intersection of All Three Sets (A ∩ B ∩ C):
   Look for the elements that are common to all three sets (A, B, and C).
   Place these elements in the central region where all three circles overlap.
   
3. Fill in the Intersections of Two Sets Only:
   A ∩ B (only): Find the elements that are in both A and B, but not in C. These go in the overlapping region of circles A and B, excluding the part that also overlaps with C.
   B ∩ C (only): Find the elements that are in both B and C, but not in A. These go in the overlapping region of circles B and C, excluding the part that also overlaps with A.
   A ∩ C (only): Find the elements that are in both A and C, but not in B. These go in the overlapping region of circles A and C, excluding the part that also overlaps with B.
   
4. Fill in the Regions for Each Set Only:
   A only: Find the elements that are in set A but are not in B and not in C. These go in the part of circle A that does not overlap with circles B or C.
   B only: Find the elements that are in set B but are not in A and not in C. These go in the part of circle B that does not overlap with circles A or C.
   C only: Find the elements that are in set C but are not in A and not in B. These go in the part of circle C that does not overlap with circles A or B.
   
5. Fill in Elements Outside All Three Sets (within U):
   Look for any elements that are part of the universal set (U) but are not listed in sets A, B, or C.
   Place these elements in the region of the rectangle that is outside all three circles.

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

How To Shade Regions Of Venn Diagrams Involving Three Sets

Venn Diagrams: Shading Regions with Three Sets, Part 1 of 2
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

• Show Video Lesson

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'

• Show Video Lesson

How To Write An Expression For A Venn Diagram Region?
Create an expression to represent the outlines part of the Venn Diagram shown.

• Show Video Lesson

How To Solve Word Problems With 3-Set Venn Diagrams?

Venn Diagram Problem With 3 Circles
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?
   
2. How many signed up only for an English class?
   
3. How many signed up for math or English?
   
4. 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?

• Show Video Lesson

How To Solve A Venn Diagram Problem Involving Three Sets?
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?

• Show Video Lesson

How to fill up a 3-circle Venn Diagram?
In this video we go over a basic word problem involving three sets. We use a Venn diagram to answer the series of questions.

• Show Video Lesson

Venn Diagram Shading Calculator Or Solver
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.

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