[6] VENN DIAGRAMS
1) One useful way of illustrating relationships among sets is by
using Venn Diagrams. The Universal set is represented by a
rectangle and each set is represented by a circle completely inside
the rectangle. The figures below show Venn Diagrams.
U
U
AA
A
A’
Set A in its Universal set.
The complement of A: A’
2) Set Operations Using Venn Diagrams
a) A ∪ B
b) A ∩ B
U
c) Disjoint sets
U
A
B
A
d) A – B (or A ∩ B’)
(complement of B in A)
U
B
A
B
e) B – A (or A’ ∩ B)
(complement of A in B)
U
A
B
A
B
f) (A ∪ B)’
g) (A ∩ B)’
U
h) A ⊆ B
U
U
B
A
B
A
B
A
3) Express the shaded region in terms of the correct set operations.
Assume that A, B represent nonempty sets.
a)
b)
A
B
d)
c)
A
B
e)
U
A
f)
U
A
B
g)
U
A
B
h)
U
B
A
B
A
B
i)
U
A
B
U
A
B
4) Use set operations to describe the interior regions of Venn diagrams.
a) The Universal set (1 region)
b) One set (2 regions)
U
U
A
I
I
c) Two sets (4 regions)
U
A
B
d) Three sets (8 regions)
U
II
I
II
A
I
II
V
III
IV
B
III
IV
VI
C
VII
VIII
5) The number of interior regions in a Venn diagram with n finite
n
sets is given by 2 , where n is the number of sets.
6) Use a Venn Diagram to shade the regions that correspond to the
indicated set operations:
b) (A – B)’
c) A ∪ B’
Two Circles: a) A’ ∩ B
d) (A’ – B)’
e) (A’ ∪ B)’
f) A’ – B’
g) A – B
h) A – B’
i) (A – B)’
j) (A ∩ B’) ∪ B
k) (A ∪ B)’ ∩ A
Three Circles: a)
c)
e)
g)
(A ∪ B) ∩ C
(A ∪ B’) ∩ C’
(A ∪ B)’ ∩ C
(A ∩ B)’ ∩ C
b) A’ ∩ (B ∪ C)
d) A ∪ (B ∩ C)’
f) A’ ∪ (B’∩ C)
h) A’ ∪ (B ∪ C)’
8) In a class of 52 students, 28 are taking a math course, 19 are
taking a chemistry course, and 10 are taking both a math course
and a chemistry course.
a) How many students are taking
U
math only?
M
C
b) How many students are taking
Neither course?
c) How many are taking exactly one
of the two courses?
d) How many are taking at least one
of the two courses?
e) How many are taking at most one of the two courses?
f) How many are taking chemistry but not mathematics?
g) How many are taking mathematics or chemistry?
9) At a particular local bank, 17% of the loans are processed by Miss
Alvarez alone, 37% by Miss Jones only, and 13% by neither one
of these two bank officers.
a) What percent of the loans are processed by Miss Alvarez and
Miss Jones?
b) What percent of the loans are processed by Miss Alvarez?
c) What percent of the loans are not processed by Miss Jones?
d) What percent of the loans are processed by at least one of the
two bank officers?
e) What percent of the loans are processed by at most one of the
two bank officers?
f) What percent of the loans are processed by exactly one of the
two bank officers?
10) In a group of 100 individuals, 37 enjoy golf, 42 enjoy fishing, 45
enjoy playing tennis, 24 enjoy golf and fishing, 19 enjoy fishing and
tennis, 21 enjoy tennis and golf, and 15 enjoy all three sports. Let G
represent the set of people who enjoy playing golf; F represents
those who enjoy fishing; T represents those who enjoy tennis.
U
a) Construct the correct Venn diagram
G
F
to summarize the information.
b) How many people enjoy none of
the three sports?
c) How many enjoy fishing only?
d) How many enjoy golf and tennis
but not fishing?
T
e) How many enjoy exactly one sport?
f) How many enjoy at most one sport?
g) How many enjoy at least one sport?
h) How many enjoyed exactly two of the three sports?
i) How many enjoyed at most two of the three sports?
j) How many enjoyed at least two of the three sports?
11) A survey of 250 people showed that 55 enjoyed water skiing only, 127
enjoyed sailing, 43 enjoyed skiing and sailing but not swimming, 82
enjoyed skiing and sailing, 66 enjoyed sailing and swimming, 21
enjoyed skiing and swimming but not sailing, and 7 enjoyed none of
the three water sports.
a) How many people enjoyed all three water sports? b) How many
enjoyed swimming only? c) How many did not enjoy swimming?
d) How many enjoyed exactly one of the three sports? e) How many
enjoyed at most one of the sports? f) How many enjoyed at least
one? g) How many enjoyed exactly two? h) How many enjoyed at
most two? i) How many enjoyed at least two? j) How many enjoyed
sailing and swimming, but not skiing? k) How many enjoyed
neither skiing nor sailing?
Exercises
1) Express the shaded regions in terms of the correct set operations.
a)
b)
c)
U
U
U
A
B
d)
A
B
e)
U
B
B
A
B
f)
U
A
A
U
A
B
2) Compute the number of interior regions in a Venn diagram
containing 10 sets.
3) Sketch a Venn Diagram like the one shown and shade the regions
that correspond to the indicated set operations
U A
B
I II III
IV
a) A’ ∪ B
b) (B – A)’
c) A ∩ B’
d) B – A
e) A’ – B
f) A – B’
g) (A – B’)’ h) (A’ ∩ B)’
i) (A’ – B’)’
h) (A ∩ B)’ ∩ A
j) (A ∩ B’) ∩ B
4) Sketch a 3-circle Venn Diagram and shade the regions that
correspond to the indicated set operations:
a) (A ∩ B) ∪ C
b) A’ ∪ (B ∩ C)
c) (A ∩ B’) ∪ C’
d) A ∩ (B ∪ C)’
e) (A ∩ B)’ ∪ C
f) A’ ∩ (B’ ∪ C)
g) (A ∪ B)’ ∪ C
h) A’ ∩ (B ∩ C)’
i) A’ ∪ (B ∩ C’)
5) In a group of 28 individuals, 9 enjoy drinking coffee, 10 enjoy drinking
tea, 2 enjoy both coffee and tea. (a) How many of these individuals
enjoy drinking coffee or tea? (b) How many enjoy tea but not coffee?
(c) How many enjoy coffee only? (d) How many do not enjoy any of
the two beverages? (e) How many enjoy exactly one of the two
beverages? (f) How many enjoy at most one of the two beverages?
(g) How many enjoy at least one of the two beverages?
6) In a survey of 65 musicians, it was found that 40 played the guitar, 13
played the piano only, 36 played either guitar only or piano only, and
25 did not play the guitar. (a) How many of the musicians played the
guitar or the piano? (b) How many played both the guitar and the
piano? c) How many played the guitar only? d) How many played
exactly one of the two instruments? e) How many played at least one
of the two instruments? f) How many played at most one of the two
instruments? g) How many played neither one of the two instruments?
7) A survey of 500 individuals revealed that 295 enjoyed reading Times
Magazine, 166 enjoyed reading The Wall Street Journal, 300 enjoyed
reading Business Week, 111 enjoyed Times Magazine and The Wall
Street Journal, 89 enjoyed The Wall Street Journal and Business
Week, 171 enjoyed Business Week and Times Magazine, and 71
enjoyed reading all three magazines.
a) How many enjoyed reading none of the three magazines?
b) How many enjoyed reading exactly one of the three magazines?
c) How many enjoyed reading exactly two of the three magazines?
d) How many enjoyed reading at least one of the three magazines?
e) How many enjoyed reading at least two of the three magazines?
f) How many enjoyed reading at most one of the three magazines?
g) How many enjoyed reading at most two of the three magazines?
h) How many enjoyed reading Times Magazine and Business Week
but not The Wall Street Journal?
i) How many enjoyed reading Business Week only?
j) How many enjoyed Times Magazine or The Wall Street Journal?
k) How many enjoyed neither Business Week nor Time Magazine?
ANSWERS TO SELECTED EXERCISES
1) a) (B – A)’ b) (A ∩ B)’ c) U d) A – B e) A f) [(A – B) ∪ (B
– A)]
2)
= 1024
3) a) shade I, III, IV b) shade I, II, III c) shade II d) shade I, II, IV e)
shade I, II, III f) shade I, II, III g) shade IV h) shade II i) shade I
j) shade none (empty set)
4) a) shade III, V, VI, VII, VIII b) I, IV, VI, VII, VIII c) I, II, III, IV, V
d) shade I
e) shade I, II, IV, V, VI, VII, VIII
f) shade I, VII, VIII
g) shade I, V, VI, VII, VIII
h) shade I, IV, VIII
5) a) 17
b) 8
c) 7
d) 11
e) 15
f) 26
g) 17
6) a) 53
b) 17
c) 23
d) 36
e) 53
f) 48
g) 12
7) a) 39
b) 232 c) 158 d) 461
e) 229
f) 271
g) 429
h) 100
i) 111 j) 350
k) 76