Math 216  Exam 2  (150 points)                             Name _____________________

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 Reference Sheet), compute:

 

 

 

 

 

 

 

 

 

d.      Using the formula for the sum of a geometric sequence (on the Reference Sheet), compute:

 

=

 

Notation: P (X) is the power set of X.

 

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 AC, the complement of A? (3 points)

a.       { x Î Z | x is odd}

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

c.       { x Î R | x = 2n + 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.      2n

c.       n2

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 A, B, and C, if A Í C and B Í C, then A È B Í C.    

 

Proof:  Suppose A, B, and C are any sets such that A Í C and B Í C.  Let x Î ___________.  [We must show that _____________.]  By the definition of union, we know that either  x Î A or _______________.  If x Î A, then we know that x Î C, since A Í C.  Similarly, if _____________, then we know that _____________, since _____________.  Therefore, ____________ [as was to be shown].

 

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 f(x) is one-to-one.  (5 points)

 

 

 

 

 

 

 

 

 

 

 

 

19.   Use mathematical induction to prove the following claim:  (9 points)

 

Claim:  1 + 3 + 5 + … + (2n – 1) = n2  for all integers n ³ 1.

 

Proof (by mathematical induction):