MATH 214 Mathematical Problem Solving for MCED
Spring 2011
Venn Diagrams1
The English logician John Venn (1834-1923)2 invented a simple visual way of describing relationships
between sets. His diagrams, now called Venn diagrams, use circles (or other planar regions) to
represent sets. Venn diagrams are fairly intuitive and best learned through examples.
Consider the sets of squares and quadrilaterals. Because every member of the set squares is also
a member of the set quadrilaterals, the set squares is a subset of the set quadrilaterals. This is
represented by a Venn diagram in which a circle denoting squares is drawn inside a circle denoting
quadrilaterals; both are enclosed by a rectangle representing the universal set of polygons, of which
both squares and quadrilaterals are subsets. Note that the size of the circles is irrelevant.
• The interior of the circle labeled squares represents the set of all squares.
• The region outside of the squares circle but inside the quadrilaterals circle represents the set
of all quadrilaterals that are not squares.
• The region outside of the quadrilaterals circle represents all geometric shapes that are not
quadrilaterals, such as hexagons, cubes, and circles.
Now consider the sets dogs and cats. A pet can be either a dog or a cat, but never both. We can
draw a Venn diagram to represent these sets by using a pair of circles that do not touch, recognizing
that the sets dogs and cats are disjoint sets. Again, we enclose the circles in a rectangle. This time,
context suggests that the region represented by the rectangle is the universe of discourse consisting
of pets.
1 Based
largely on material from Using and Understanding Mathematics: a quantitative reasoning approach, 5th
ed., by Jeffrey Bennett and William Briggs, Addison-Wesley, 2011.
2 John Venn was an Anglican priest who wrote extensively on logic, statistics, and probability theory. He also wrote
a history of Gonville and Caius College at Cambridge University, where he was both a student and teacher. Though
well respected for many of his contributions to logic, he is best known for the diagrams that bear his name.
For our last general case, consider the sets nurses and women. As shown in the figure below, these
are represented by overlapping circles because it is possible for a person to be both a woman and a
nurse. Because of the overlapping, this diagram displays four regions:
• The overlapping region represents people who are both women and nurses, namely female
nurses.
• The non-overlapping region on the nurses circle represents nurses who are not women – that
is, male nurses.
• The non-overlapping region on the women circle represents women who are not nurses.
• From context, we interpret the region outside both circles to represent people who are neither
nurses nor women – that is, men who are not nurses.
Note that the sizes of the regions are not important. For instance, the smaller relative size of the
overlapping region does not imply that female nurses are less common than male nurses. In fact, it
is possible that some of the regions may represent sets with no members at all. In other words, we
use overlapping circles whenever two sets might have members in common.
Example 1: Describe the relationship between the given pairs of sets and draw a Venn diagram
showing this relationship. The universal set is described in parentheses. Interpret all the regions of
the Venn diagram.
a. Democrats and Republicans (registered voters in the United States)
b. Nobel Prize winners and Pulitzer Prize winners (people)
2
Solution 1:
a. A voter can be registered for only one political party, so the sets Democrats and Republicans
are disjoint. The region outside both circles represents voters who are neither Democrats nor
Republicans – that is, Independents, or voters registered with some other party.
b. It is possible for a person to win both a Nobel Prize and a Pulitzer Prize, so these are overlapping sets, even though the overlap contains very few people.3 The region outside both circles
represents people who have not won either prize (which is most of humanity!).
Example 2: Draw a Venn diagram showing the relationships amongst the sets of natural numbers,
whole numbers, integers, rational numbers, and real numbers. Where would irrational numbers be
found in this diagram? Soluion 2: Because each of these sets is a subset in turn of the next set
listed, the Venn diagram is structured with nested circles. Irrational numbers are real numbers that
are not rational, so they are found inside the outer circle of real numbers but outside the circle of
rational numbers. 3 In fact, Pearl Buck (1892-1973) and Toni Morrison (1931-), both American writers, are recipients of a Nobel Prize
for Literature as well as a Pulitzer Prize for Fiction, Buck in 1932 and 1938 and Morrison in 1988 and 1993.
3
Categorical Propositions
The proposition all squares are quadrilaterals makes a very specific claim–namely, that the set squares
is a subset of the set quadrilaterals. Propositions of this type are called categorical propositions
because they claim a particular relationship between two categories or sets.
Like all propositions, categorical propositions must have the structure of a complete sentence. Categorical propositions also have another important general feature. Of the two sets in a categorical
proposition, one set appears in the subject of the sentence and the other in the predicate. For
example, in the proposition all squares are quadrilaterals the set squares is the subject set and the
set quadrilaterals is the predicate set. We will use the letter S to denote the subject set and the
letter P to denote the predicate set, so that we can rewrite all squares are quadrilaterals in the form
all S are P, where S = squares and P = quadrilaterals.
Categorical propositions come in the following four standard forms:4
The Standard Categorical Propositions
Form
All S are P
No S are P
Some S are P
Some S are not P
Example
All squares are quadrilaterals.
No circles are quadrilaterals.
Some positive integers are even.
Some rational numbers are not positive.
Subject Set S
squares
circles
positive integers
rational numbers
Predicate Set P
quadrilaterals
quadrilaterals
even numbers
positive numbers
Because categorical propositions express a relationship between two sets, we can use what we’ve
already learned about Venn diagrams to make visual representations of categorical propositions.
However, in the case of overlapping sets, we must be specific in stating which parts of the overlap
are addressed by the proposition. The following example should this idea clear.
Example 3: Draw a Venn diagram to represent each of the four sample propositions in the table
above.
Solution 3: The proposition all squares are quadrilaterals claims that the set S = squares is a subset
of P = quadrilaterals. The diagram is pictured below. Note that all propositions of the form All S
are P would have the same basic Venn diagram.
4 The four forms listed in this table are sometimes called by logicians universal positive, universal negative, existential positive, and existential negative propositions, respectively.
4
The proposition no circles are quadrilaterals tells us that the set S = circles is disjoint from the set
P = quadrilaterals. That is, they have no common members. The Venn diagram therefore shows
two separated circles. The Venn diagram for the proposition some positive integers are even requires
overlapping circles for S = positive integers and P = even numbers. However, we must be specific
about claim made in the proposition. In this case, the proposition asserts that there are some
numbers (at least one) which are both positive integers and even numbers. To highlight this fact we
place an X in the overlapping region of the diagram. Note that the proposition does not specifically
indicate that there exist positive integers which are not even, nor does it assert that there are even
numbers which are positive (even though both facts are clearly true). The given proposition does
not claim that the other regions in the diagram contain any members. The proposition some rational
numbers are not positive also requires overlapping circles. This proposition asserts that there are
some rational numbers (at least one) who are in the S = rational numbers circle but not in the
P = positive numbers circle. Therefore, we put an X in the non-overlapping region of the rational
numbers circle. No claim is made about whether there are rational numbers which are also positive
or positive numbers which are not rational (even though there certainly do exist such numbers). 5
Many statements make claims about relationships between two categories, but don’t look precisely
like one of the four standard forms for categorical propositions. It’s often useful to rephrase such
statements in one of the standard forms. For example, the statement every triangle with two congruent base angles is isosceles can be rephrased to read all triangles with two congruent base angles
are isosceles triangles. Now the statement has the form All S are P, where S = triangles with two
congruent base angles and P = isosceles triangles.
Example 4: Rephrase each of the following statements in one of the four standard forms for categorical propositions.
a. Some quadratic equations can be solved by factoring.
b. Whole numbers whose ones digit is 7 are never divisible by 5.
Solution 4:
a. Some quadratic equations can be solved by factoring can be rephrased as some quadratic equations are equations that can be solved by factoring. It now has the form some S are P where
S = quadratic equations and P = equations that can be solved by factoring. The Venn diagram
is pictured below, with an X in the overlapping region to indicate the claim that there is at
least one quadratic equation that can be solved by factoring.
b. Whole numbers whose ones digit is 7 are never divisible by 5 can be rephrased as no whole
numbers whose ones digit is 7 are whole numbers divisible by 5. This proposition has the form
no S are P, where S = whole numbers whose ones digit is 7 and P = whole numbers divisible
by 5.
6
Venn diagrams are particularly useful for dealing with three sets that may overlap one another. For
example, suppose you are conducting a statistical study to learn how teenage employment rates
differ between boys and girls and between honor students and others. For each teenager in your
study, you need to record the answers to these three questions:
• Is the teenager a boy or girl?
• Is the teenager an honor student or not?
• Is the teenager employed or not?
The diagram below can help you organize this information. Notice that it has three circles, representing the sets boys, honor students, and employed persons. Because the circles overlap one another,
they form a total of eight regions (including the region outside all three circles). This diagram also
has labels for the other five regions, which would not normally be included, so as not to complicate
the diagram, but which have been added here to clarify the interpretation of each region. Also,
observe that the same information could be recorded in a diagram representing the three sets girls,
honor students, and employed persons, or boys, honor students, and unemployed persons.
Example 5: You hire an assistant to help you with the study of teenage employment described above.
He focuses on a small group of teenagers who are all enrolled in the same school. He reports the
following facts about this group:
• Some of the boys who are honor students are unemployed.
• Some of the girls who are not honor students are employed.
7
Place X’s in the appropriate places in the diagram to indicate the regions that you can be sure have
members. Based on this report, do you know whether there are any of the school’s honor students
who are unemployed girls? Why or why not?
Solution 5: The diagram is pictured below with X’s placed in the proper regions. The region
corresponding to the school’s honor students who are unemployed girls is the region in the upper
right of the honor students circle, and since there is no X in this region, the given information
provides no guarantee that this region has any members. Therefore, among the teenagers at this
school, we do not know whether there are any honor students who are unemployed girls. Other Uses of Venn Diagrams
So far we have used Venn diagrams only to describe relationships, such as whether two sets overlap
or whether overlapping sets share common members. Venn diagrams can be even more useful when
we add specific information, such as the number of members in each set or overlapping region.
Consider the study summarized in the table below, an example of what is called a two-way (or
contingency) table. This study was designed to learn whether a pregnant mother’s status as a
smoker or nonsmoker affects whether she delivers a low or normal birth weight baby.
Mother’s
smoking status
Smoker
Nonsmoker
Baby’s birth weight
Low
Normal
18
132
14
186
The table records counts, which correspond to the four combinations of the baby’s birth weight
status and the mother’s smoking status.
Although a two-way table is a good way to summarize data, a pictorial representation may be easier
to understand. Because there are four combinations of birth weight status and smoking status, we
can use a Venn diagram to provide such a picture, using two overlapping circles to display four
regions. One way of building the diagram is shown below, the circles representing the sets smoking
mothers and low birth weight babies. This diagram differs from others we have shown only in that
the regions are now labeled by how many members each contains, the counts coming from the table
8
above. The Venn diagram5 makes it easy to to see how smoking affected babies in the study. Notice
that normal birth weight babies were much more common than low birth weight babies among
both smokers and nonsmokers. However the smoking mothers had a lower proportion of normal
birth weight babies and a higher proportion of low birth weight babies. This suggests that smoking
increases the risk of having a low birth weight baby, a fact that has been borne out by careful
statistical analysis of this and other studies.
Example 6: Human blood is often classified according to whether three antigens, A, B and Rh, are
present or absent. Blood type is stated first in terms of the A and B antigens: blood containing
only A is called type A, blood containing only B is called type B, blood containing both types A
and B is called type AB, and blood containing neither antigen is called type O. The presence or
absence of the antigen Rh is indicated by adding the word positive (present) or negative (absent),
or the corresponding symbol ‘+’ or ‘−.’ The table below shows the eight resulting blood types and
the rough percentage of people with each type in the U.S. population. Draw a Venn diagram to
illustrate these data.
Blood type
A positive
B positive
%
34
8
Blood type
A negative
B negative
%
8
2
Blood type
AB positive
O positive
%
3
35
Blood type
AB negative
O negative
%
1
9
Solution 6: We can think of the three antigens as three sets A, B, and Rh, and draw a Venn diagram
with three overlapping circles. We do this below. Note the eight regions, each labeled with its type
and percentage of the population. 5 Do
you see that a similar Venn diagram could be constructed using the sets nonsmoking mothers and normal
birth weight babies?
9
Exercises
Use set notation to write the members of the following sets.
1. Every third number between 4 and 20, beginning with 4
2. All perfect squares between 30 and 130
3. All triangles with five sides
4. All rational numbers whose denominator is three times its numerator
Draw Venn diagrams for each of the following groups of sets. Describe the members of each region
in the diagram, or state that a region contains no members.
5. Negative integers and natural numbers
6. Multiples of 4 and multiples of 5
7. Whole numbers the sum of whose digits equals 5, prime numbers, and numbers less than 100
8. Convex6 polygons, regular polygons, and polygons with at least one obtuse interior angle
For the given categorical propositions, Draw a Venn diagram displaying the subject and predicate
sets. Rephrase the given statement if necessary.
9. All widows are women.
10. Some movie stars are redheads.
11. No negative number can have a real square root.
12. Not every quadratic equation can be solved by factoring.
Draw Venn diagrams that represent the following situations. Use only the diagram (and no other
information) to give answers to the questions that follow.
13. All multiples of 6 are even numbers. No even numbers have ones digit 5. Some even numbers
are multiples of 4. No numbers with ones digit 5 are between 7 and 13. Questions: Could
there be multiples of 6 whose ones digit is 5? Could there be multiples of 6 which are multiples
of 4? Are there even numbers between 7 and 13? Are there multiples of 6 between 7 and 13?
14. All meat has protein. All dairy products have protein. Some beans have protein. All beans,
but no meat or dairy products, are plants. Questions: Could there be beans that are dairy
products? Could there be meat that is a dairy product? Could there be dairy products that
are plants? Could there be plants with protein?
Draw Venn diagrams that represent the following situations. Mark each region with the correct
number of members.
15. One hundred people who grew up in either New York City or Los Angeles were surveyed to
determine whether they preferred hip-hop music or rock music (both and neither were not
acceptable responses). Of those who grew up in Los Angeles, 20 preferred hip-hop and 40
preferred rock. Of those who grew up in New York, 30 preferred hip-hop.
6A
convex polygon is one for which every line segment joining a pair of vertices remains inside or on the boundary
of the polygon. For instance, a five-sided polygon in the traditional shape of a house is convex whereas a polygon in
the shape of a star of David is not.
10
16. In a trial of a new allergy medicine, 120 people were given the medicine and 80 were given a
placebo. Of those given the medicine, 90 showed improvement in their allergy symptoms. Of
those give the placebo, 30 did not show any improvement.
17. A survey of 130 individuals revealed the following results about the news sources that they
regularly used.
TV/radio only
Internet only
Newspapers only
None
20
29
15
6
TV/radio and internet only
TV/radio and newspapers only
Internet and newspapers only
All three sources
12
18
22
8
18. The popular vote received by Barak Obama in the 2008 presidential election can be divided
according to gender and party as follows (vote counts, in millions, are approximate).
Democrats
Republicans
Independents and Others
11
Women
23.8
2.2
10.3
Men
21.1
1.9
9.2