Dr. Jaffe
Advanced Mathematics
Homework due Tuesday 10/27
NAME: _______________________________________________________
1. Name an infinite set and two infinite subsets of that set.
S = {x | x is an integer}
Subset = {x | x is a positive integer}
Subset = {x | x is a negative integer}
2. If you have a set with an order of 9, what are two ways to determine the number of subsets of that set?
We can do 29. When we add a new element to the set we are doubling the number of subsets because we add that
new element to every existing set.
We can do 9C0 + 9C1 + 9C2 + 9C3 + 9C4 + 9C5 + 9C6 + 9C7 + 9C8 + 9C9. We are getting all the number of subsets with
0 elements by choosing 0 elements from the set, then we get the number of subsets with only one element by
choosing one element from the set. We get the number of subsets with 2,3,4,5,6,7,8, and 9 elements in the same way.
Then we can add all of those together to get the total number of subsets.
3. What role do combinations play in determining the number of subsets? Why?
We are determining the number of subsets with r elements by choosing that many elements from the set; nCr, where n
= the number of elements in the set and r = the number of elements in the subset. The order does not matter so we
use combinations.
4. Why do we use combinations to determine the number of subsets versus permutations?
If we used permutations then it would be saying (a, b) is different from (b, a), but in sets order does not matter, so
we use combinations.
5. Name a set that you see in the world around you? Why do we group these things?
An example would be the email groups we set up. The elements of the set consist of the people in the class. We
group these people so we can send out a mass email to all the people in the class as opposed to having to type in up
to 34 names individually.
6. Name two separate sets in the world around you (one of them can be the one you used in question 5) that
you can combine to be one set.
We can have the set of classes and the set of people in a class.
7. How many proper subsets are there of the people in this class?
213 - 1. We subtract one because we do not want to include the subset that includes all the people in this class. That
is not a proper subset.
Dr. Jaffe
Advanced Mathematics
Homework due Wednesday 10/28
NAME: ______________________________________________________
1. Can you conclude A = B if A and B are two sets with the same power set?
Yes, the power set is the list of all subsets. If all the subsets are the same, then all the elements in
A and B must be the same.
2. Suppose that A = {2, 4, 6}, B = {2, 6}, C = {4, 6} and D = {4, 6, 8}. Determine which of
these sets are subsets of which other of these sets.
B  A (all the elements in B are in A)
C  D (all the elements in C are in D)
3. How many different elements does A x B have if A has m elements and B has n
elements? Explain your answer.
Every element in A is grouped with every element in B so the total number of eleements in A x B
is m*n
4. Let A = {a, b, c}, B = {x, y}, and C = {0, 1}. Find
a) A x B x C
{(a,x,0), (a,x,1), (a,y,0), (a,y,1), (b,x,0), (b,x,1), (b,y,0), (b,y,1), (c,x,0), (c,x,1), (c,y,0),
(c,y,1)}
b) C x B x A
{(0,x,a), (0,x,b), (0,x,c), (0,y,a), (0,y,b), (0,y,c), (1,x,a), (1,x,b), (1,x,c), (1,y,a), (1,y,b),
(1,y,c)}
c) C x A x B
{(0,a,x), (0,a,y), (0,b,x), (0,b,y), (0,c,x), (0,c,y), (1,a,x), (1,a,y), (1,b,x), (1,b,y), (1,c,x),
(1,c,y)}
d) B x B x B
{(x,x,x), (x,x,y), (x,y,x), (x,y,y), (y,x,x), (y,x,y), (y,y,x), (y,y,y)}
5. Suppose that A x B = ∅, where A and B are sets. What can you conclude?
A and B must both be the empty set.
6. What is the Cartesian product of A x B, where A is the set of courses offered by the
mathematics department at a university and B is the set of mathematics professors at this
university?
A x B would be all the possible classes the ways we can combine a math professor and the
courses they teach.
7. Suppose that A, B, and C are sets such that A ⊆ B and B ⊆ C. Show that A ⊆ C.
If A ⊆ B, then all the elements that are in A are also in B. If B ⊆ C, then all the elements in B,
which includes all the elements in A are also in C. This means all the elements in A are in C.
Dr. Jaffe
Advanced Mathematics
Homework due Thursday 10/29
NAME: ______________________________________________________
(From the Packet)
1.
a) {6} These are the elements that are in A and B
b) {2,3,4,6,8,9,10} These are the elements that are in A and the elements that are in B
c){1,3,9} These are the elements in C that are not in A
d) {1,3,4,5,6,7,9} These are the elements that are not the set A and C
e){2,3,8,9,10} These are the elements that are in A or B and then also in C
f) A All the elements that are in A or B and also in A include only the elements in A
2.
a) T The empty set is a subset of every set
b) F If the elements are not in A then they cannot also be in A
c) T The elements in the complement of A are the elements not in A. If we take the
not in that set, then we have the elements that are in A
3.
a) {1,3,4,5,7,9} These are all the elements that are in the set of A or in the set of B
b){1,5,9} These are all the elements that are in both sets
c)
2 6,8,10
3,7
4
1, 5, 9
4.
a) {3,12}
b){6}
c){3,8,10}
d){3,12}
e){6}
f) all integers except 6
g) {(3,6),(3,8),(3,10),(6,6),(6,8),(6,10)}
5. (A  B)' = A'  B'
You can draw the Venn Diagrams for both and they look as follows
elements that are
6.
a)
A
B
C
b)
A
B
C
7. {··· ,−4,−3,−2,−1,0,1,2}
8. 26= 63
9.
a) (A−B) ∪ (B−A)
b) (A∪B)' ∪ (A∩B)
c) B−A
d) A−(A∩B∩C)
e) [(A∩B)∪(A∩C)∪(B∩A)]−[A∩B∩C]
f) [(A∪B)−(A∩B)]−(B∩C)
10.
M
E
19
16
8
7
11
10
12
H
n[(M∩E) ∪ (M∩H) ∪ (E∩H)] = 36 This will allow us to find the number of people in (E ∩H), which was not given
15 + 18 + (7 + x) - 7 - 7 = 36 We subtract 7 twice because we counted three times (in each intersection)
15 + 11 + x = 36
x = 10
b) 19
11. .
N
W
17
7
12
10
9
10
6
PS
b) 90 - (17 + 7 + 6 + 12 + 10 + 9 + 10) = 19
12. 18+6−4 = 20
13.
a) {A♥,K♥}, {A♥,K♥,Q♥}, {A♥,K♥,J♥}, {A♥,K♥,Q♥,J♥}.
b) 250.
14.
a) True
b) False
c) True
d) True
e) False
f) True
g) True
h) False
i) True
j) True
k) False