Venn Diagrams Part 2 2012
1. More coloring for credit! Note: you may want to print out this page. For each set description, shade the corresponding portion
of the Venn diagram:
a) A
b) B '
A
A
B
B
U
A
A
A B C
A
C
B
A
B
A
U
B
U
U
C
B
A  B'
A
B
k) A  ( B  C )
A B C
U
h)
U
j)
B
U
g) ( A  B )'
U
i)
A
B
U
B
A B
d)
A
f) ( A  B )'
A' B'
e)
A B
c)
A
l) A  ( B  C )
C
B
U
A
U
C
B
U
2. Matching. In a particular school, some students study Spanish (S), some study French (F), some study both, and some study
neither. Match each written phrase with its corresponding set. (Note: there is a one-to-one
F
S
correspondence between questions and answers, so each question has exactly one distinct
matching answer.)
a) The set of students who study Spanish.
i) ( S  F )'
d) The set of students who study neither language.
SF
iii) F  S '
iv) ( F  S ' )  ( S  F ' )
e) The set of students who study both languages.
v)
b) The set of students who do not study Spanish.
c) The set of students who study at least one language.
ii)
f) The set of students who study only French.
U
vi) ( S  F )'
g) The set of students who study exactly one language.
(vii)
h) The set of students in the school.
S
(viii) n(F )
i) The set of students who do not study French.
(ix)
j) The set of students who are not studying both languages.
k) The number of students who are studying French.
S'
(x) F '
(xi) S  F
U
3. For each region of the Venn diagram, a region or regions have been indicated with a dot. Write an expression to describe the
region. One has been done for you.
Example:
A
a)
b)
A
B
B
U
answer: A  B' (the region
that is in A, but not in B.)
d)
A
B
A
e)
C
B
U
A
C
B
U
g)
A
U
C
B
U
f)
A
U
c)
C
B
A
U
C
B
U
4. Consider the following situation: In a class of 40 students, 25 of them are studying French, and 18 of them are studying
Spanish. Eight are studying neither language.
a) Make a Venn diagram representing this situation. Remember that the numbers 40, 25, and 18 will not directly appear
on your Venn diagram, since there is some overlap. (Some students study both languages.)
b) Determine the probability that a randomly-selected student is studying French.
c) Determine the probability that a randomly-selected student is studying exactly one language.
d) Determine the probability that a randomly-selected student is studying both languages.
e) Determine the probability that a randomly-selected student is studying French, if it is known that this student is
studying Spanish.
5. Consider the Venn diagram at right, which summarizes the results of a sports survey, in which
T
12 4 3 S
respondents indicated whether they enjoyed Tennis, Soccer, or Football.
a) How many people were surveyed?
3 5 7
b) Overall, which sport was most popular?
9
15
F
c) Suppose we know a particular respondent enjoys Tennis: What is the probability this
U
person enjoys Soccer?
d) We write P(Soccer | Tennis) to indicate the probability that a person enjoys Soccer,
given that they enjoy Tennis. What would P(Tennis | Soccer) mean?
e) Calculate P(Soccer | Tennis).
f) Calculate P(Tennis | Soccer).
g) Which is greater: P(Soccer | Tennis) or P(Tennis | Soccer)?
h) Suppose it is known that a respondent enjoys exactly two of the three sports: what is the probability this person enjoys
Football?
6. Suppose we begin with the universal set as the whole numbers between 0 and 10 inclusive. Some of these numbers are placed
into set A , some into set B , and some into set
C . Suppose we know that set A is {2,3,5,6} .
A  B is {2,3} . B  C is {0,2,7} .
A B C ?
b) Suppose furthermore that we know A  C is {2,6} , but A  C is {0,2,3,5,6,7,8} . What whole numbers can we
be certain are not members of set B ?
a) What do we know about