Venn Diagrams
A Venn diagram is a diagram that shows all possible logical relations between a finite
collection of different sets. They can be used to illustrate simple set relationships in probability,
logic, and statistics.
Before we can effectively use a Venn diagram, we must first understand Sets.
Think of a Set as a well-defined collection of objects or elements in which is it possible to
determine if a given object is included in the collection. Sets are often labeled with a letter such
as A = {1, 2, 3}.
Example: U = {1,2,3,4,5,6,7,8,9, 10, 11}, A = {1,3,5,7} and B = {2,4,5,7,9,11}
Find each set.
(a) A ∩ B
Solution: Set A ∩ B contains the elements that are found in both set A and B. Or in other words,
their intersection.
A ∩ B = {5,7}
Now that we understand how a set works, let’s apply this knowledge to the use of a Venn
diagram.
Example:
A researcher collecting data on 100 households finds that
76 have cats;
21 have dogs; and
12 have both dogs and cats.
The researcher wants to answer the following questions.
(a) How many do not have a cat?
(b) How many have neither a cat nor a dog?
(c) How many have a dog but not a cat?
Solution: A Venn diagram will help us sort out this information. First, put the number 12 in the
region common to both a cat and a dog, because 12 households have both.
1
U
Cats
12
Dogs
Of the 21 with a dog, 21 – 12 = 9 households with no cat, so we put 9 in the region for a dog but
no cat. Similarly, 76 – 12 = 64 households with a cat but not a dog, so we put 64 in that region.
Finally, the diagram shows that 100 – 64 – 12 – 9 = 15 households who have neither a dog nor
a cat. Now we can answer the questions.
(a) 15 + 9 = 24 do not have a cat.
(b) 15 have neither.
(c) 9 have a dog but not a cat.
U = universal set (all households surveyed)
Cats
64
12
Dogs
9
15
2