What is discussed in this lecture?

1.   Examples of Three-Set Venn Diagrams
2.   The Basic Three-Set Venn Diagram

1.   Examples of Three-Set Venn Diagrams

In the previous lesson we studied two-set Venn diagrams.  Now, we will expand our knowledge to three-set Venn diagrams.  There are many real world multi-set Venn diagrams that I found by searching the web.  Venn diagrams are used in the sciences, in languages, in social studies, and in economics.

Example 1:

The following Venn Diagram shows types of English words that are different in spelling and meanings.  Image source from http://www.wired.com.

Example 2:

The following Venn Diagram shows types of English words that are different in spelling and meanings.  Image source from http://images-mediawiki-sites.thefullwiki.org.

Example 3:

We can also use Venn diagrams to show the relationships in large organizations such as the European Economic Union.  Image source from http://craphound.com.

2.   The Basic Three-Set Venn Diagram

Now, that we have seen some examples of multi-set Venn diagrams, let's look at a basic three-set Venn diagram.

NOTE:  1 = I     2 = II     3 = III     4 = IV     5 = V     6 = VI     7 = VII     8 = VIII

Please note that the regions in the above Venn diagram are again identified by Roman numerals.  The numbering is arbitrary, that is, we can number any region as I, any region as II, and so on.  What is usually done in mathematics, is to start the numbers in the upper left-hand corner and ending with VIII as the area outside the three sets as shown.

Region I - The elements that belong to set A, but not to set B and set C.

Region II - The elements that belong to set A and to set B, but not to set C.

Region III - The elements that belong to set B, but not to set A and set C.

Region IV - The elements that belong to set A and set C, but not to set B.

Region V - The elements that belong to sets A, B, and C.

Region VI - The elements that belong to set B and set C, but not to set A.

Region VII - The elements that belong to set C, but not to set A and set B.

Region VIII - The elements that belong to the universal set U that are not in sets A, B, or C.

Now we are going to look at two examples that combine complements, unions, and intersections of three sets.  We will use the information in three-set Venn Diagram to answer some questions.

Example 4:

Use the following Venn Diagram to answer each question.

a.  Which regions represent set B' ?

This is the complement of set B.  That is, we must list all of the regions that are NOT in set B.

I, IV, VII, VIII

b.  Which regions represent set B ?

II, III, V, VI

c.  Which regions represent set ?

This is the union of set A and set C.

I, II, VI, V, VI, VII

d.  Which regions represents set ?

This is the intersection of set A and set B.

II, V

e.  Which regions represent set ?

This is the intersection of set A, set B, and set C.

V

f.  Which regions represent set ?

I, II, III, IV, V, VI, VII

g.  Which regions represent set ?

This is the complement of set .  That is, we must list all of the regions that are NOT in set .

VIII

Example 5:

Use the following Venn Diagram to answer each question.

a.  Which elements are in set B' ?  Write your answer in roster form!

This is the complement of set B.  That is, we must list all of the regions that are NOT in set B.

{Alex, Ingrid, Karin, Rita, Mike, Ron, Carol, Bill, Justin, Leanne, Margie}

b.  Which elements are in set B ?  Write your answer in roster form!

{Kay, George, Steve, Beverly, Ben}

c.  Which elements are in set ?  Write your answer in roster form!

This is the union of set A and set C.

{Alex, Ingrid, Karin, Kay, Rita, Ben, Mike, Ron, Carol}

d.  Which elements are in set ?  Write your answer in roster form!

This is the intersection of set A and set B.

{Kay, Ben}

e.  Which elements are regions represent set ?

This is in set A, set B, and set C.  Write your answer in roster form!

{Ben}

Example 6:

A teacher is working with a group of students which she has classified by whether or not they scored 85% or above on each of three exams.  The result are shown in the Venn Diagram below.  Use the Venn Diagram to write in roster form the set of students who scored 85% or above on Exam 1 OR not on Exam 2.

You are asked to represent the set of students who scored 85% or above on Exam 1 OR not on Exam 2.

In a previous lecture we discussed the various combinations of sets.  We talked about complements (not), intersections (and), and unions or).   Furthermore, "but" means the same thing as "and". Thus, "but" means intersection.

In the given problem, we primarily have an OR situation, that is, we are dealing with a union. The first part are the students who took Exam 1.  The second part of the union consists of the compliment (NOT) of Exam 2, which consists of all of the students outside of the Exam 2 set.

All students who scored 85% or above on Exam 1 are

     Alex, Ingrid, Karin, Kay, Ben, Rita

All students who are outside of the set of Exam 2 are

     Ingrid, Karin, Rita, Mike, Carol, Ron, Bill, Justin, Leanne, Margie

Finally, we find the union of these two sets (remember, we have an OR situation):

    {Alex, Ingrid, Karin, Kay, Ben, Rita, Mike, Carol, Ron, Bill, Justin, Leanne, Margie}

That is, we have to list ALL names, but only once. Incidentally, if you had been asked for an intersection you would have only listed the names that the two sets have in common.

Example 5:

Examine the following three-set Venn Diagram and then answer some questions.

a.  Find the set of students who scored 85% or above on Exam 2.

{Kay, Ben, George, Steve, Beverly, Ron, Rita}

b.  Find the set of students who scored 85% or above on Exam 3.

{Ben, Ron, Rita, Mike, Carol}

c.  Find the set of students who scored 85% or above on Exam 1 AND on Exam 3 (intersection).

{Ben}

d.  Find the set of students who scored 85% or above on Exam 1 AND on Exam 2 (intersection).

{Kay, Ben}

e.  Find the set of students who scored 85% or above on Exam 1 OR on Exam 3 (union).

{Ingrid, Karin, Ben, Kay, Rita, Ron, Mike, Carol}

f.  Find the set of students who scored 85% or above on Exam 1 OR on Exam 2 (union).

{Ingrid, Karin, Kay, Ben, Ron, Rita, George, Steve, Beverly}

g.  Find the set of students who scored 90% or above on Exam 1 AND NOT on Exam 2 (intersection).

Here we must find the intersection of circle 1 with the compliment of circle 2.  Remember, the complement of set A, symbolized by A', is the set of all elements in the universal set that are NOT in set A!

{Ingrid, Karin}

h.  Find the set of students who scored 85% or above on Exam 3 AND NOT on Exam 1 (intersection).

Here we must find the intersection of circle 3 with the compliment of circle 1. 

{Mike, Carol, Ron, Rita}

i.  Find the set of students who scored 85% or above on Exam 1 OR NOT on Exam 2 (union).

Here we must find the union of circle 1 with the compliment of circle 2.  That is, list all names that are in circle 1 in addition to all names that are outside circle 2.

{Ingrid, Karin, Kay, Ben, Mike, Carol, Bill, Justin, Leanne, Margie, Alex}

j.  Find the set of students who scored 85% or above on Exam 3 OR NOT on Exam 1 (union).

Here we must find the union of circle 1 with the compliment of circle 2.  That is, list all names that are in circle 1 in addition to all names that are outside circle 2.

{Ben, Ron, Rita, Mike, Carol, George, Steve, Beverly, Bill, Justin, Leanne, Margie, Alex}

k.  Find the set of students who scored 85% or above on exactly one test.

{Ingrid, Karin, George, Steve, Beverly, Mike, Carol}

l.  Find the set of students who scored 85% or above on at least two tests.

{Kay, Ben, Ron, Rita}

m.  Find the set of students who scored 85% or above on Exam 2 AND NOT on Exam 1 AND NOT on Exam 3 (intersection).

Here we must find the intersection of circle 2 with the compliment of circle 1 and the compliment of circle 3. 

{George, Steve, Beverly}

n.  Find the set of students who scored 85% or above on Exam 1 AND NOT on Exam 2 AND NOT on Exam 3 (intersection).

Here we must find the intersection of circle 2 with the compliment of circle 1 and the compliment of circle 3. 

{Ingrid, Karin}

o.  Find the set of students who scored less than 85% on all exams.

{Bill, Justin, Leanne, Margie, Alex}