Chapter 2
Sets and Functions
Section 2.2
Operations on Two Sets
As we have seen before when we were talking about logic the real interesting and
useful things happen when two categories of things are being discussed at once.
We will give a more detailed way to show how two categories of things can interact.
I want to start by going back to a previous example.
Let:
J = Months that begin with the letter J = {January, June, July}
Y = Months that end with the letter Y = {January, February, May, July}
U = Universal Set (All months) = {Jan, Feb, Mar, Apr, May, Jun, Jul, Aug, Sep, Oct, Nov, Dec}
Intersection
Union
The intersection of two sets is marked
with the symbol . It represents the
elements that are common to both
sets at the same time.
The union of two sets is marked with
the symbol . It represents what you
get by putting together the elements
in both sets (i.e. the set containing all
the elements in both sets).
J  Y = {January, July}
J  Y = {January, February, May, June, July}
Venn Diagrams
To represent two different sets in a Venn Diagram we put all the elements in the
diagram (only once!)in the correct region for the set(s) they are in.
For example
J = {January, June, July}
Y = {January, February, May, July}
U = {January, February, March, April, May, June, July, August, September, October, November, December}
U
March
J
Y
January
May
August
June
July
April
February
November
December
September
October
J = {May, February, March, April, August, September October, November, December}
Y = {June, March, April, August, September October, November, December}
Various regions in the Venn Diagram can be referred to symbolically by combining
the set operations intersection (), union () and complement (¯).
U
March
J
Y
January
May
August
June
July
April
February
November
December
September
October
J Y
= Elements in J and at the same time not in Y
= {June}
J Y
= Elements not in J and at the same time in Y
= {May, February}
J Y
= Elements in neither J nor Y
= {March, April, August, September, October, November, December}
Numerical Information About Sets
We can construct the Venn Diagrams to include the number of elements in various
components of sets. Rather than listing all the elements in a set we would just put in
the number in that region.
U
A
B
horse
mouse
dog
bird
cat
rat
pig
U
A
frog
B
snake
fish
3
2
2
cow
4
The number in set A is: 5
The number in set B is: 4
The number in AB is: 2
(This is how many are in both A and B.)
The number in AB is: 7
(This is how many are in either A or B.)
The number in
A  B is: 3 (This is how many are in A but not in B (i.e. just A).)
The number in A  B is: 4 (This is how many are in neither A nor B)
Numerical Problems with Sets
Given some numerical information about sets of things and how they are related
we can extract more information based on what we know about sets.
Consider the following problem:
A survey of 100 people at the DMV asked them if they owned a car or a truck. The
results were that 78 people said they owned a car and 46 people said they owned
a truck. Of the 46 people who said they owned a truck 29 said they also owned a
car. Answer each of the following questions:
How many own both a truck and car? 29
How many own either a truck or a car? 95
The idea here is to think in terms of sets
and fill in the corresponding numbers in the
Venn Diagram.
How many own just a truck? 17
Let the set C = Car Owners
How many own just a car? 49
Let the set T = Truck Owners
How many own neither a truck nor a car? 5
How many do not own a car? 22
How many do not own a truck? 54
U
C
T
49
29
17
5
Numerical Information in Tables
Another way that information about the number of items in a set can be collected is
in the form of a table. Lets look at the following problem.
A survey is taken of 200 students in a dorm asking if they were: male, female,
democrat, republican or independent. The results are given in the table below.
Democrat
(D)
Republican
(R)
Independent
(I)
Female (F)
54
32
7
Male (M)
41
46
20
Each category is represented by a set.
D = Set of Democrats
F = Set of Females
R = Set of Republicans
M = Set of Males
I = Set of Independents
We can ask some more complicated questions about these categories of people.
Democrat
(D)
Republican
(R)
Independent
(I)
Female (F)
54
32
7
Male (M)
41
46
20
1. How many males are there?
(i.e. The number in the set M)
41 + 46 + 20 = 107
2. How many Democrats are there?
(i.e. The number in the set D)
54 + 41 = 95
3. How many are both male and democrat?
(i.e. The number in the set MD)
41
4. How many are either female or independent?
(i.e. The number in the set FI)
54 + 32 + 7 + 20 = 113
5. How many are not republican?
54 + 41 + 7 + 20 = 122
(i.e. The number in the set R )