Math 216  Exam 2  (150 points)                             Name _____________________

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 B? (4 points)

a.       {6, 10, 12, 14, 18, 24}

b.      {12}

c.       {6, 18, 24}

d.     

 

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.     

 

4.      Which of the following is false? (4 points)

a.       3 {1, 2, 3, 4, 5}

b.      3 {1, 2, 3, 4, 5}

c.       {2, 4} {1, 2, 3, 4, 5}

d.      {a, t, e} {e, a, t}

 

5.      Let A = {x Z | x is odd} and the universal set U = Z, the set of all integers.  Which of the following is AC, the complement of A? (4 points)

a.       { x Z | x is even}

b.      { x R | x = 2n for some n R}

c.       { x Z | x = 2n + 1 for some n Z}

d.      all of the above

 

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

a.       A B

b.      A B = A

c.       A P (B)

d.      A 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.      2n

c.       n2

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.      2n

c.       n2

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 (B 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 B .

 

                                    A =

                                    B =

                                    C =

 

 

12.  Fill in the blank.  (5 points)

 

Claim:  For all sets A, B, and C, if A B and B C, then A C.   

 

Proof:  Suppose A, B, and C are any sets such that A B and B C.  To show that A C, we must show that every element in ___________ is in ___________.  But given any element in A, that element is in ญญญญญญญญญญญญญญญญญญญญญญญญญญญ_______________ (because A 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)

 

______________________________

ญญญญญญญญญญญญญญญญญญญญญญญญญ______________________________

______________________________

______________________________

______________________________

______________________________

______________________________

______________________________

 

 

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 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.

 

 

 

 

 

 

18.  For the following functions, indicate if it is one-to-one and/or onto.  (4 points)

 

S = {a, b, c}   

Z = all integers (includes positive, negative, and 0)  

function definitions

one-to-one?

(yes-no)

onto?

(yes-no)

f:SS and is the set of ordered pairs {(a,b), (b,a), (c,c)}

 

 

f:ZZ 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 0.  Prove that f(x) is one-to-one.  (4 points)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

21.  Let f: R R with  for all real numbers x 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 = n2 + n,  for all integers n 1.

 

Proof (by mathematical induction):