Download SET THEORY

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SET THEORY
I. Basic Notions
A set is a collection of objects. The objects in the set are referred to as elements. A set can be
finite, meaning that there is a limited number of objects in the set or infinite, meaning that there
is no end to the number of objects in the set.
We usually use a capital letter to designate a set. There are two ways that we can specify a set.
One way is called a roster, in which we list the elements of the set in set brackets:
A = { a,b,c,d,e} or M = { 1,2,3,…9,10} or C = { 1,2,3,4,…}
The “…” part of set M does not mean it is infinite; it simply means that a pattern has been
established and the reader can fill in the missing numbers. The “…” portion of set C, however,
does mean that it is infinite, no numbers are specified to end the list of numbers. All three of
these sets have been designated by roster.
The other way to specify a set is using set-builder notation. This notation includes the set
brackets, a variable to indicate a member of the set and the a vertical bar (read as “such that”)
and then a description of the elements of the set. Examples are:
B = { x| x is a real number and -2 < x < 4 } or D = { y| y is a state in the USA}
Set B is an infinite set and D is a finite set but both are specified by set-builder notation.
For a set to be useful, it must be well-defined. For instance, sets B and D above are well-defined.
It is easy for us to determine if a particular object is a member of the set. An example of a set
that is not well defined would be {x| x is Susie’s friend}—we do not know how well a person
needs to know Susie to be classified as a “friend.”
There is a special symbol used to indicate the relationship “is an element of”.
The following statement is read “ a is an element of A”
a A
I. EXERCISES
Find the elements in each set. List all elements in a roster.
1) { 11, 12, 13, …,16}
2) { 8,9,10,…16}
3) { x| x is a counting number less than 7}
4) { x| x is a counting number greater than 4 but less than 10}
5) { x| x is an integer and -3 ≤ x < 5}
6) { x| x is a vowel }
7) { 17, 22, 27, …, 47}
8) { 74, 68, 62, …, 38}
9) { x| x is an odd counting number between 8 and 16}
10) {x| x is an odd counting number not greater than 15}
11) {x| x is an even counting number less than 20}
12) {x| x is a counting number less than 100 and is evenly divisible by 21}
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Identify each set as finite or infinite.
13) { 3,4,5,…8}
14) {1, ½ , ⅓, ¼, …}
15) {0, 1,2,3,…,75}
16) {x| x is a state in the United States}
17) {s| x is a real number }
18) {x| x is a letter in the English alphabet}
19) {x| x is an integer and -5 <x < 3}
20) {x| x is a rational number and 0 < x < 1}
Identify each set as well-defined or not well-defined.
21) { 3,4,5}
22) { x| x is a member of this year’s football team at your school}
23) { 5, 10, 15, 20, …}
24) { x| x is a real number greater than 2}
25) { x| x is a rational number}
26) { x| x is a good movie}
27) { x| x is a lousy artist}
28) { x| x is a counting number less than 1}
29) { x| x is a mouse weighing over 500 pounds}
Fill in the blank with either
30)
31)
32)
33)
34)
35)
 or  to make a true statement.
3 ______ { x| x is an odd counting number}
-4______ { x| x is an integer and |x| > 5}
0 ______ {x| x is a counting number }
{3}_____ { 1, 2, 3, 4, 5 }
-12_____{ 10, 12, 14, 16}
{12}____{ 10,12,14,16}
Write TRUE or FALSE for each of the following.
36)3  {1,2,3}
37)15  {0,5,10,15,20,...}
37)5  {0,2,4,6,...}
38)1  {7,5,3,1}
39){2}  {0,2,4,6}
40)  1  {2,4,6,8}
Write TRUE of FALSE for each statement, using the following sets:
A = {2, 4, 6, 8, 10} B = { 2, 4, 8} C = { 4, 6, 12 }
41) 4  A
42)8  C
43)0  B
44) 2  C
45)6  A
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II. Subsets
For a set to be useful, it must be well-defined. For instance, sets B and D above are well-defined.
It is easy for us to determine if a particular object is a member of the set. An example of a set
that is not well defined would be {x| x is Susie’s friend}—we do not know how well a person
needs to know Susie to be classified as a “friend.”
There is a special symbol used to indicate the relationship “is an element of”.
The following statement is read “ a is an element of A”
a A
Sets can have four relationships. They are:
1) A=B Equal sets contain exactly the same elements, in any order.
2) Sets A and B are disjoint. Disjoint sets have no elements in common.
3) Sets A and B overlap. This means that there are some elements the two sets have in
common, but there are other elements that belong to just A and some that belong to just B.
4) A is a subset of B. This means that every element in A is also in B. The following
statement is read “ A is a subset of B.”
A B
Every set is considered to be a subset of itself. Also, there is a special set, called the null or
empty set, the set that does not contain any elements at all. This set is indicated by set brackets
with nothing in between them, { }, or by a special symbol, Ø. The empty set is considered to be
a subset of every set also. For each subset with n elements, there are 2n subsets.
Another useful set to consider is the universal set, the set containing all the elements under
current consideration. The universal set is usually indicated by the letter U. The set that is
considered the universal set might change from problem to problem.
Once a universal set is defined, it is possible to also define the complement of a set. The
complement of set A, denoted as A', is the set of all elements in U that are not in A.
For example if U = { 1,2,3,4,5} and A = { 1,3,5}, then A' = { 2,4}.
II. EXERCISES
Insert the symbol
1)
2)
3)
4)
5)
6)
7)
8)
9)

or

to make a true statement.
{1,3}_______{1,3,5,7}
{5}_______{x| x is an integer and 0 < x< 5}
{ } _______{ 6,7,8}
{3,4}_______{3,4}
{1,2}_______{x| x is a counting number}
4______{2,4,6,8,…}
{4}______{2,3,4,5,…}
{ } _____{ }
{3,4,5,6}________{x| x is an odd counting number}
Identify each statement as TRUE or FALSE, using the following sets.
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A= {2,4,6,8,10, 12} B={2,4,6}
C ={4,10,12}
D = {2,10} E = { 2,4,6,8,10,12,14}
10) A  B
11) B  A
12) B  B
13){}  A
14)C  B
15) D  A
16) D  E
17)C  D
18) D  C
29) B  E
20) There are exactly 8 subsets of C.
21) There are exactly 4 subsets of D.
22) There are exactly 6 subsets of B.
23) There is only 1 subset of Ø.
Find the number of elements in each set. Then find the number of subsets for each set.
24) { 4,5,6}
25) { 6}
26) {0,1,2,3}
27) { }
28) {x| x is a counting number and 5 < x< 10}
29) {x| x is an integer and |x| ≤ 3}
Let U = { 0,1,2,3,4,5,6,7,8}. Find the complement of each set.
30) {2,4,6}
31) {x| x is an even counting number less than 10}
32) {5,6,7,8}
33) {3}
34) { }
35) U
Let C represent the set of symptoms for a cold. Then C = { sneezing, red nose, aches}
Let F represent the set of symptoms for flu. Then F = {sneezing, red nose, sore throat,
fever}
Let U represent the set containing the combined symptoms for cold and flu.
List the elements for the following sets:
36) U
37) C'
38) F'
aches,
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III. Set Operations
There are two set operations that allow us to form new sets. Those operations are
Union, in which we combine all members of both sets into a new set (but we do not repeat
common elements) and intersection, in which we list the common elements for the two sets.
Some examples
may make this clear. We will use these symbols:
 means int er sec tion
 means union
A = { 1,2,3,4,5} and B = { 2,4,6} and C = {1,2,3,4,5,6}
A  B  {2,4}
A  C  {1,2,3,4,5}
A  B  {1,2,3,4,5,6}
We can also combine the notion of complements with the idea of union and intersection.
Let U = {1,2,3,4,5,6,7,8} and sets A, B and C be as above. Then
A' B  {6,7,8}  {2,4,6}  {6}
( A  B)'  {1,2,3,4,5,6}'  {7,8}
( B  C )  A'  {2,4,6}  {6,7,8}  {2,4,6,7,8}
III. EXERCISES
Write TRUE or FALSE for each statement.
1){1,3,5,7}  {5,6,7,8}  {5,7}
2){2,1,5}  {1,2,3}  {1,2,3,4,5}
3){3,5,7}  {1,2,3,4,5}  {3,5}
4){}  {8,7,6,5}  {8,7,6,5}
5){}  {8,7,6,5}  {8,7,6,5}
6){1,3,4}  {1,3,4}  {}
7){1,3,4}  {1,3,4}  {1,3,4}
8){}  {}  {}
9){}  {}  {}
10){2,4}  {3,4,5}  {2,3,4,5}
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For the next set of exercises, use these sets:
U={3,4,5,6,7,8,9}
X={3,4,5}
Z={4,7,8,9}
Y={3,5,7,8}
List the elements in each set:
11) X  Y
12) X  Y
13)Y  Z
14)Y  Z
15) X  U
16) Z  U
17) X '
18)Y 1'
19) X 'Y '
20) X ' Z
21) Z '{}
22)Y '{}
23) X  (Y  Z )
24)Y  ( X  Z )
25)(Y  Z ' )  X
26)( X 'Y '0  Z
27)( X  Y )' Z '
28)(Y ' X )' Z
For the following problems, describe each set in words:
A={all students over 17}
B={all students studying Discrete Math}
C={all students studying psychology} D={all students with brown eyes}
U={all students}
29) A  B
30) B  C
31) A  D
32) A  C
33) A'C
34) A' B '
35)( A') B  C
36)( A  B )  C '
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For the following statements, assume sets A and B are any two sets. Classify the statements as
TRUE or FALSE.
37) A  ( A  B )
38) A  ( A  B )
39)( A  B )  A
40)( A  B )  A
For the following exercises, let sets X, Y, Z and U be defined as in exercises 1-28. Find the set
defined on the left hand side of the equals sign and then find the set defined on the right hand
side. If the two are the same, write TRUE. If they are different, write FALSE.
41) X  Y  Y  X
42)( X ' )'  X
43) X  {}  {}
44)Y  {}  Y
45) Z  {}  Z
46)( X  Y )  Z  X  (Y  Z )
47( X  Y )  Z  X  (Y  Z )
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IV. Venn Diagrams
We can depict set relationships in pictures called Venn diagrams. In these pictures, the universal
set is seen as a rectangle and two sets are seen as overlapping circles. We shade in the parts of
the picture that represent the set under consideration. Consider the following pictures:
IV. EXERCISES
Place the elements of these sets in the proper location in the Venn diagram.
1) U=a,b,c,d,e,f,g}
A={b,d,f,g}
B={a,b,d,e,g}
2) U={5,6,7,8,9,10,11}
A={5,8,9,10}
B={7,8,9,10}
Use a Venn diagram similar to the ones above to shade each of the following sets.
3) B  A'
4) A  B
5) A' B
6) A' B '
7) B ' A
8) A' A
9)U '
10)( A  B )  B '
11) A'( A  B )
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Venn diagrams can also be used to depict relationships among three sets.
12)Use these sets to place the elements in the correct spot in the Venn diagram.
U={1,2,3,4,5,6,7,8,9}
A=(1,3,5,6}
B=(2,4,6,8}
C={2,3,4,5,6}
Use a Venn diagram to shade each of the following sets.
13)( A  B)  C
14)( A  B)  C
15))( A' B)  C
16)( A  B)'C
17)( A'C ' )  B
18)( A  B' )  C '
19)( A  C ' )  B'
20) B  ( A  C ' )'
Venn diagrams can be used to solve counting problems. In the following problem, use the
information given to count the number of people who fit into each region of a Venn diagram
with three circles. Start with the most involved area (the intersection of all 3 sets). Don’t forget
to draw the rectangle for the universal set.
Suppose a survey produces the following results:
120 take French
150 take History
115 take Science
63 take French and History
70 take History and Science
78 take French and Science
38 take all three
6 students do not take any of these three courses
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We can then answer questions like :
a) How many students take only History?
b) How many students take only French?
c) How many students were in this survey?
Use Venn diagrams to answer the following:
21) A survey of 150 students concerning their club memberships at school showed that
78 students were in a foreign language club
60 students were in an honor club
54 students were in a service club
25 students were in foreign language and honor clubs
30 students were in honor and service clubs
32 students were in service and foreign language clubs
11 students were in all 3 types of clubs
a)
b)
c)
d)
How many students were only in an honor club?
How many students were only in a foreign language club?
How many students were only in a service club?
How many students were not in any of these three clubs?
22) Athletes were questioned as to their participation in three sports. Here are the results:
34 played soccer
61 played basketball
50 played tennis
25 played soccer and tennis
30 played basketball and tennis
8 played soccer and tennis
6 played all three
9 played none of these sports
a)
b)
c)
d)
How many athletes played only basketball?
How many played only tennis?
How many played only one sport?
How many athletes were questioned?
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A B
A B
A'
A' B