Critical Thinking and Problem Solving

Fourth Edition, second revision - September 1998

Donald Nute
Department of Philosophy
The University of Georgia

Copyright (c) 1997, 1998 Donald Nute

^M Unit 3: Venn Diagrams

Constructing Venn Diagrams
Summary of Principles for Making Venn Diagrams
Summary of Principles for Reading Venn Diagrams
Exercises on Venn Diagrams
Practice Quiz 1 - Interactive
Practice Quiz 1 - Solutions
Practice Quiz 4 - Interactive
Practice Quiz 4 - Solutions
Practice Quiz 5 - Interactive
Practice Quiz 5 - Solutions

Unit 3: Venn Diagrams

Constructing Venn Diagrams

A Venn diagram provides a way to represent information about individuals and groups. In English, some of the kinds of information conveyed by Venn diagrams is expressed using quantifier terms such as `all', `some', and `none'. We use regions labelled with capital letters to represent groups and we use lowercase letters to represent individuals. Together, these components give us a visual representation of the relations we are interested in. By constructing a diagram that represents some initial set of relations, we can often immediately deduce other relations from the diagram. These include relations between groups as well as membership of particular individuals in groups.

Figure VENN-1

The basic form of the Venn diagram is three intersecting circles inside a box. We have an example of this basic form in Figure VENN-1. In this diagram, each of the circles represents a group of individuals or objects that have some common property or characteristic. We can think of the circles as a fence drawn around these individuals. Each of the numbered regions formed by the intersection of the circles also represents a group. We normally label the circles with a capital letter that stands for the features the things in the circle have in common. Let A stand for Albanians, B stand for barbers, and C stand for cello players. Then in Figure VENN-1, anything in regions 1, 4, 6, and 7 is an Albanian, anything in regions 4 and 7 is an Albanian barber, and anything in region 7 is an Albanian barber that plays the cello. Notice that region 8 outside all the circles is where we would find those individuals that are not Albanians, barbers, or cello players.

Figure VENN-2

We show that a region is empty by shading it. For example, in Figure VENN-2 we have shaded regions 1 and 6 to show that they are empty. What exactly does this mean in terms of our three properties? If regions 1 and 6 are empty, then anything in circle A must be in regions 4 and 7. But regions 4 and 7 are also inside circle B. So anything in circle A must also be in circle B. Or to put is another way, all Albanians are barbers.

This gives us a method for representing the information given in an English sentence of the form `All S are P', where S is the subject of the sentence and P is the predicate. We draw a Venn diagram and label one of the circles S and another P. Then we shade the part of the circle labeled S that is outside the circle labeled P. Sometimes, though, the subject of the English sentence is not represented by an entire circle. Take for example the sentence `All Albanian barbers play the cello'. There is no circle in our diagram representing Albanian barbers. Instead, they are represented by the two regions 4 and 7. What the English sentence says is that everything in these two regions is a cello player, i.e., is inside circle C. But region 4 is outside circle C. So to show that all Albanian barbers play the cello, we must shade region 4 as in Figure VENN-3. In general, then, to show that all S are P, we identify the region in the diagram corresponding to S (call this the S-region) and shade any part of this region that is outside the P-region.

Figure VENN-3

Figure VENN-4

What information do we represent if we shade regions 4 and 7 as in Figure VENN-4? It tells us that the intersection of the Albanian circle and the barber circle is empty. So no Albanians are barbers. Conversely, no barbers are Albanians. How would we then represent the information that no Albanian barbers are cello players? We would find the region that represents the Albanian barbers (i.e., the football-shaped region made up of regions 4 and 7) and shade the portion of that region where cello players might be found (i.e., the part inside circle C). In general, to represent an English sentence of the form `No S are P', we shade the intersection of the S-region and the P-region.

Figure VENN-5

People often find the word `only' confusing. Consider the sentence `Only citizens can vote'. How would we represent this in a Venn diagram? The first idea most people have is that we shade the part of the citizen-region outside the voter-region. Look at Figure VENN-5 where C stands for citizens, V for voters, and M for minors (in this case, persons under 18). If we follow our first instinct and shade all of the C-region outside the V-region, what we end up saying is that all citizens can vote, including minor citizens. There is no unshaded part of the minor-citizen-region outside the voter-region. Something has gone wrong. It's true that only citizens can vote, but it certainly isn't true that all citizens (including minors) can vote. We have represented something that we did not intend to represent.

Figure VENN-6

Let's think about this further. If only citizens are voters, then all voters must be citizens. What we want to do, then, is shade all of the voter-region that is outside the citizen-region as in Figure VENN-6. And in general, to represent `Only S are P', you should shade all of the P-region outside the S-region. Perhaps the safest way to remember this is just to reverse an `only' and change it to an `all' whenever you see it: `Only S are P' is diagrammed the same way as `All P are S'.

Shading, then, is the operation we use to represent the information that some region of a Venn diagram is empty. But how do we show that a region is occupied? We will use letters to represent individuals and put those letters in the regions they occupy. Since we are using capital letters to represent properties or features of things, we will use lower case letters to represent individuals. Sometimes we will know something about a particular individual. We might know, for example, that George is a barber who plays the cello. We can use the letter g to represent George in a diagram. Since we know who g stands for, we say that it is a constant. Other times, we may only know that something or other is in a region, for example, that there must be Albanian barbers although we don't know who they are. Then we will use an x, y, or z as a variable that stands for one of these unknown Albanian barbers.

To show that George is a cello-playing barber, we need to put a g in every part of the cello-playing-barber-region, i.e., in every part of the intersection of circles B and C. So we put a g in regions 5 and 7. To show that George is a single individual, we link the two g's with a curved line called a bridge. We also want to show that something is an Albanian barber; so we put x's in regions 4 and 7 and connect them with a bridge. The result is Figure VENN-7.

Figure VENN-7

From Figure VENN-7, we can't tell whether George is Albanian. He could be in either region 5 or 7. If he is in region 7, he is Albanian. But if he is in region 5, he is not. Suppose we learn that George and Ellen are not Albanian. We represent this new information about George by blotting out the g in region 7. But we don't have any e's in the diagram; so we need to find another way to show that Ellen is not Albanian. We do this by putting an e in each region outside circle A, i.e., in regions 2, 3, 5, and 8. Again, we connect these e's with bridges. The result of these changes is shown in Figure VENN-8.

Figure VENN-8

We show that a region in a Venn diagram is empty, then, by shading that region. And we show that individuals either have or do not have various properties and features by adding constants or variables to the diagram or blotting out constants and variables. We can also read information from a Venn diagram. We find out that English sentences using the quantifiers `all', `no', and `only' are true by noticing which regions of the diagram are shaded, and we find out that sentences using names or the quantifiers `some' and `not all' are true by noticing where constants and variables are located in the diagram.

Look again at Figure VENN-8. If we learn that no Albanians play the cello, we will shade the intersection of circles A and C. This gives Figure VENN-9 which represents the following information:

1. George is a cello-playing barber.
2. Some Albanians are barbers.
3. George and Ellen are not Albanians.
4. No Albanians play the cello.

Figure VENN-9

But the diagram gives us more information. Notice that we have an x in the barber-region but no x in any unshaded part of the cello-player-region. That means that there is someone who is a barber that does not play the cello, i.e., that some barbers do not play the cello. Can we also conclude that some cello players are not barbers? We see that there is an e in the cello-player-region 3, which is outside the barber-region. But there are also e's inside the barber region. We know that Ellen is in region 2, 3, 5, or 8, but we can't locate her any better than that. So we cannot conclude that some cell players are not barbers.

On the next two pages, we have a summary of these and other principles for representing information in Venn diagrams and for reading information from Venn diagrams. The only way to really get a feel for how we can use this method to form new conclusions from the information we have is by practicing. After the summaries of principles, you will find some exercises and practice quizzes. Notice that we can use a Venn diagram either to test whether an argument is reliable or to extract new information from the information we already have. So Venn diagrams are both a tool for evaluating arguments and a tool for problem solving.
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Summary of Principles for Making Venn Diagrams

Summary of Principles for Reading Venn Diagrams

Exercises on Venn Diagram

Use Venn diagrams to determine whether the grounds guarantee the claims in the following arguments. To do this, represent the grounds in the Venn diagram, then see if the diagram guarantees the claim. Solutions.

1. Only dragons breathe fire, and all dragons fly. So all flying dragons breathe fire.
2. Some sea creatures are mammals, and some mammals have well-developed social structures. So some sea creatures have well-developed social structures.
3. All flying mammals eat insects. Some mammals do not eat insects. So some mammals do not fly.
4. Bats are flying mammals. The Red Baron is a flying mammal. So the Red Baron is a bat.
5. Penguins are found only in Antarctica. No cichlasomas are found in Antarctica. So penguins are not cichlasomas.
6. All professional athletes are overpaid. Bruce Jenner is an athlete, but he is not a professional. So he is not overpaid.
7. No genius is tidy. Some children are geniuses. So some children are not tidy.
8. Only hard-workers pass this course. Some of my students are hard workers. So some of my students will pass this course.
9. None but the beautiful may join the club. I am not beautiful. All the snobs are in the club. So I am not a snob.
10. Tom and Dick are both singers, but only Tom sings opera. Only opera singers sing at the Metropolitan, and even some opera singers don't sing at the Metropolitan. So Dick doesn't sing at the Metropolitan.
11. All rich people drive big cars. A Cadillac is a big car. So Jones must be rich, because he drives a Cadillac.
12. Only flying mammals have sonar. My pet is a mammal and it doesn't have sonar. So it doesn't fly.
13. Some fish care for their young. No fish that care for their young are nothobranches. So some nothobranches are not fish.
14. All beautiful actresses are famous. Jane is beautiful, but she is not an actress. So Jane is not famous.
15. Some people like okra and some don't. People who like okra are strange. So not all people are strange (i.e., some people are not strange).
16. All poor people eat at McDonald's. Some people eat at McDonald's and some don't. So some people are poor.
17. Only actors live in Los Angeles. Some people who live in Los Angeles are rich. So some actors are rich.
18. Some students goof off. No student who goofs off passes my course. So some students do not pass my course.
19. Ivan is a Russian. All Russians who are not communists are traitors. Ivan is not a traitor. So Ivan is a communist.
20. Bradley was a philosopher. All philosophers are either idealists or realist. Bradley was not a realist, so he was an idealist.
21. Not all kangaroos are grey. Grey kangaroos are not endangered. So not all kangaroos are endangered.
22. All elephants and rhinos are pachyderms. Some pachyderms are not rhinos. So some rhinos are not elephants.
23. Some mushrooms are non-poisonous. Only death's cap mushrooms are poisonous. So some mushrooms are not death's cap mushrooms.