Plus homework points (max of
50) ________
1. Questions on sequences,
sums, and products. (4 points each)
a. Write the first four terms of the
following sequence:
_{}
b. Give an explicit formula for the sequence whose initial terms are as
follows:
_{}
c. Compute the following summation:
_{}
d. Using the formula for the
sum of an arithmetic sequence
(on the Reference Sheet), compute:
_{}
e. Using the formula for the
sum of a geometric sequence
(on the Reference Sheet),
compute:
_{}=
Notation: P
(X) is the power set of X; Z is the set of all integers; R
is the set of all real numbers.
Multiple Choice: Circle the letter of the best answer to each
of the following:
2. Let A = {6, 12, 18, 24} and
B = {10, 12, 14}. Which of the following
is A Ç B? (4 points)
a. {6, 10, 12, 14, 18, 24}
b. {12}
c. {6, 18, 24}
d. Æ
3. Let A = {6, 12, 18, 24} and
B = {10, 12, 14}. Which of the following
is A – B? (4 points)
a. {6, 10, 12, 14, 18, 24}
b. {12}
c. {6, 18, 24}
d. Æ
4. Which of the following is
false? (4 points)
a. 3 Î {1, 2, 3, 4, 5}
b. 3 Í {1, 2, 3, 4, 5}
c. {2, 4} Í {1, 2, 3, 4, 5}
d. {a, t, e} Í {e, a, t}
5. Let A = {x Î Z  x is odd} and the
universal set U = Z, the set of all
integers. Which of the following is A^{C},
the complement of A? (4 points)
a. { x Î Z  x is even}
b. { x Î R  x = 2n for some n Î R}
c. { x Î Z  x = 2n + 1 for some n Î Z}
d. all of the above
6. Let A = {2, 4, 8, 12} and B
= {x Î Z  x is even}. Which of the following is false? (4 points)
a. A Í B
b. A Ç B = A
c. A Î P
(B)
d. A Í P
(B)
7. Let A be a set that contains
n elements. How many elements are in the set A ´ A?
(4 points)
a. 2n
b. 2^{n}
c. n^{2}
d. n!
8. Let A be a set that contains
n elements. How many elements are in P
(A), the power set of A? (4 points)
a. 2n
b. 2^{n}
c. n^{2}
d. n!
9. Find X ´ Y where X = {1, 2} and Y = {m, n}. (4 points)
10. Shade in that portion of the
Venn diagram below that represents A Ç (B È C). (4 points)
11. Disprove the following claim by providing a counterexample. (4 points)
For all sets A,
B, and C, if A is a subset of C and B is a subset of C, then A Ç B ¹ Æ.
A =
B =
C =
12. Fill in the blank. (5 points)
Claim: For all sets A, B, and C, if A Í B and B Í C,
then A Í C.
Proof: Suppose A,
B, and C are any sets such that A
Í B and B Í C.
To show that A
Í
C, we must show that every element in
___________ is in ___________.
But given any element in A, that
element is in _______________ (because A Í B), and so that
element is also in ______________ (because ___________________). Hence ___________________ [as was to be shown].
13. Using the Reference Sheet provided, supply
the appropriate reason for each step in the derivation. (8 points)
______________________________
______________________________
______________________________
______________________________
______________________________
______________________________
______________________________
______________________________
14. A couple plans to have 3
children. Let G stand for a girl and B stand
for a boy, and let, for example, GBB stand for three children where the first
is a girl (the oldest) followed by two boys.
a. List all elements of the
sample space of all possible sequences of 3 children? (4 points)
b. What is the probability that
that the couple will have 3 children of all the same sex? (4 points)
c. What is the probability that
the couple will have more boys than girls?
(4 points)
15. You must select a new
password for your eSpaceBook
account. Assume that no uppercase letters
are allowed, only lowercase. Digits are
also allowed.
a. How many different passwords
are possible if they must contain exactly 6 characters (lowercase letters or
digits)? (4 points)
b. How many different passwords
of length 6 are possible if no repetitions are allowed and the password must
end with 2 digits? (4 points)
c. How many different passwords
are possible if they may contain either 6, 7, or 8
characters (letters or digits, repetitions allowed)? (4 points)
16. What is _{} (2 points)
17. Follow the instructions
given. (3 points each)
Use arrows to draw a function that
is not onetoone.
Use arrows to draw a function that
is onetoone
but not onto.
Use arrows to draw a function that
is not onto.
18. For the following functions,
indicate if it is onetoone and/or onto. (4 points)
S = {a, b, c}
Z = all integers (includes positive, negative, and
0)
function definitions 
onetoone? (yesno) 
onto? (yesno) 
f:S®S and is the set of ordered pairs {(a,b), (b,a), (c,c)} 


f:Z®Z where f(x) = 2x 


19. Let f: R ® R with _{} , and let g: R ® R with _{}.
a. What is _{}? (4 points)
b. What is _{}? (4 points)
20. Let f: R ® R with _{} for all real numbers x
¹ 0. Prove that f(x) is onetoone. (4 points)
21. Let f: R ® R with _{} for all real numbers x
¹ 0. Prove that f(x) is not onto by providing a counterexample. (4 points)
22. How many integers from 1
through 25 must you pick in order to be sure that at least one is odd? (3 points)
23. How many cards must you pull
from a standard deck to ensure that you have at least 4 of the same suit? (3 points)
24. What is the smallest number
of people you must have to ensure that there are 3 people who were born in the
same month? (3 points)
25. In a group of 50 people,
must at least 4 have been born in the same month? (3 points)
26. Use mathematical
induction to prove
the following claim: (10 points)
Claim: 2 + 4 + 6
+ … + 2n = n^{2} + n, for all integers n ³ 1.
Proof (by mathematical induction):