1.
2.
Basic Regions
Basic Regions and Counting
1
Each Venn diagram divides the universal set into
non-overlapping regions called basic regions.
2
Show
the basic regions for the Venn diagram with
two sets and with three sets.
3
For the survey of the presidents of the 500 largest
corporations in the US where 310 had degrees
(of any sort) in business, 238 had undergraduate
degrees in business, and 184 had graduate
degrees in business:
a. Draw a Venn diagram and determine the
number of elements in each basic region;
b. Determine the number of presidents having
exactly one degree in business.
4
n(I) = n(U) - n(S T)
= 500 - 310 = 190
n(III) = n(S T) = 112
n(II) = n(S) - n(S T)
= 238 - 112 = 126
n(IV) = n(T) - n(S T)
= 184 - 112 = 72
5
The number of presidents with exactly one
business degree corresponds to the shaded region.
Adding the number of people in the two shaded
basic regions gives 126 + 72 = 198.
6
Problems involving sets of people (or other
objects) sometimes require analyzing known
information about certain subsets to obtain
cardinal numbers of other subsets. The “known
information” is often obtained by administering
a survey.
Slide 2-4-7
Suppose that a group of 140 people were questioned
about particular sports that they watch regularly and the
following information was produced.
93 like football
70 like baseball
40 like hockey
40 like football and baseball
25 like baseball and hockey
28 like football and hockey
20 like all three
a) How many people like only football?
b) How many people don’t like any of the sports?
Slide 2-4-8
Construct a Venn diagram. Let F = football,
B = baseball, and H = hockey.
B
F
20
Start with like all 3
H
Slide 2-4-9
Construct a Venn diagram. Let F = football,
B = baseball, and H = hockey.
F
B
20
8
20
Subtract to get
5
H
Slide 2-4-10
Construct a Venn diagram. Let F = football,
B = baseball, and H = hockey.
F
20
45
8
20
B
25
Subtract to get
5
7
H
Slide 2-4-11
Construct a Venn diagram. Let F = football,
B = baseball, and H = hockey.
F
20
45
8
20
7
H
B
25
Subtract total shown
from 140 to get
5
10
Slide 2-4-12
Solution
(from the Venn diagram)
a) 45 like only football
b) 10 do not like any sports
Slide 2-4-13
The universal set can be divided up into a
number of non-overlapping regions called basic
regions. These regions can be used to determine the
number of elements in a variety of subsets of the
universal set.

14