Homework points (max of 50)
________

1. Questions on sequences,
sums, and products. (3 points each)

a. Write the ** first three terms** of the
following sequence:

_{}

b. Write the following using ** summation notation**:

_{}

c. Using the formula for the
sum of an ** arithmetic sequence**
(on the

_{}

d. Using the formula for the
sum of a ** geometric sequence**
(on the

_{}=

** Notation**: P
(X) is the

** Multiple Choice**: Circle the letter of the best answer to each
of the following:

2. Let A = {2, 4, 8, 12} and B
= {2, 3, 4, 5}. Which of the following
is A Ç B? (3 points)

a. {2, 4}

b. {8, 12}

c. {3, 5, 8, 12}

d. Æ

3. Let A = {2, 4, 8, 12} and B
= {2, 3, 4, 5}. Which of the following
is A – B? (3 points)

a. {2, 4}

b. {8, 12}

c. {3, 5, 8, 12}

d. Æ

4. Let A = {*x* Î **Z** | *x* is even} and the
universal set U = **Z**, the set of all
integers. Which of the following is A^{C},
the complement of A? (3 points)

a. {* x* Î **Z** | *x* is odd}

b. {* x* Î **Z** | *x* = 2*n* for some *n* Î **Z**}

c. {* x* Î **R** | *x* = 2*n* + 1 for some *n* Î **R**}

d. all of the above

5. Let A = {2, 4, 8, 12} and B
= {*x* Î **Z** | *x* is even}. Which of the following is ** false**? (3 points)

a. A Í B

b. A Ç B = A

c. A Î P
(B)

d. A Í P
(B)

6. Let A be a set that contains
*n* elements. How many elements are in the set A ´ A? (3 points)

a. *n*

b. 2*n*

c. *n*^{2}

d. *n*!

7. Which of the following
describes the shaded portion of the Venn diagram below? (3 points)

a. A È (B Ç C)

b. (A È C) Ç B

c. (A Ç B) È C

d. C Ç (B È A)

8. Fill in the blank. (5 points)

** Claim**: For all sets

** Proof**: Suppose

9. Using the ** Reference Sheet** provided, supply
the appropriate reason for each step in the derivation. (8 points)

______________________________

______________________________

______________________________

______________________________

______________________________

______________________________

______________________________

______________________________

10. Suppose that on a true/false
exam, you have no idea at all about the answers to 3 questions. You choose answers randomly and therefore
have a 50-50 chance of being correct on any one answer. Let C stand for correct guess and W stand for
wrong guess, and let, for example, CWW stand for the case where you guessed the
first one correctly and the other two wrongly.

a. List all elements of the
sample space whose outcomes are all possible sequences of correct and wrong
guesses to these 3 questions? (3 points)

b. What is the probability that
you guessed all 3 questions correctly? (3 points)

c. What is the probability that
you guessed more questions correctly than wrongly? (3 points)

11. You must select a new
password for your *eSpaceBook* account.
Assume that no uppercase letters are
allowed, only lowercase.

a. How many different passwords
are possible if they must contain exactly 5 letters? (3 points)

b. How many different passwords
are possible if they must contain exactly 5 letters if no repetitions are
allowed? (3 points)

c. How many different passwords
are possible if they may contain either 5, 6, or 7 letters? (3 points)

12. What is _{} (2 points)

13. Follow the instructions
given. (2 points each)

Use arrows to draw a *function* that

is __not__*one-to-one*.

Use arrows to draw a *function* that

__is__*one-to-one* but __not__*onto*.

Use arrows to draw a *function* that

is __not__*onto*.

14. Let *f: R* ® *R* with _{} , and let *g: R* ® *R* with _{}.

a. What is _{}? (4 points)

b. What is _{}? (4 points)

15. What is the smallest number
of people you must have to ensure that there are 2 people who were born in the
same month? (3 points)

16. In a group of 30 people,
must at least 3 have been born in the same month? (3 points)

17. 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)

18. Let *f: R* ® *R* with _{} for all real numbers. ** Prove**
that

19. Use ** mathematical
induction** to

__Claim__: 1 + 3 + 5 +
… + (2*n* – 1) = *n*^{2} for all
integers *n* ³ 1.

__Proof__ (by mathematical induction):