1. Provide a counter example to the following statement: section 2.1 #3a

             Every geometric figures with four right angles is a square.                      (5)

 

                   A rectangle has four right angles and isn’t a square

 

  1. Prove the following:  Section 2.1 #16 minor variant

For every integer n, the number  3(n2 + 2n+ 3) – 2n2  is a perfect square.  (5)

 

                   3(n2 + 2n +3) – 2n2 =

                    3n2 + 6n + 9 – 2n2  =

                      n2 + 6n + 9 =

                      (n+3) (n+3)  =

                      (n+3) 2

  1. Prove or disprove the following statement:  Section 2.1 #47                   (5)

for n  an even integer, 2n   - 1  is not prime.

 

           for n = 3, 2n = 8. 2n – 1 = 7, 7 is prime.

  statement disproved by finding a counter example

 

  1. Prove the following statement:     section 2.1  # 27 variant  

The difference of the cubes of two consecutive integers is odd.

 

(2n+1)(2n+1)(2n+1)- (2n)(2n)(2n)

(4n2+4n+1)(2n+1) – 8n3

8n3+ 8n2 + 2n + 4n2 + 4n + 1 – 8n3

12n2 + 6n + 1

2(6n2 + 3n) + 1 

2k + 1  which is odd

 

  1. Use mathematical induction to prove that the following statement is true for every positive integer n.  Section 2.2 # 6                                              (5)

5+10+15+...+5n = [(5n)(5n+1)]/2

n = 1,     5*1 = 5      5(1)(1+1)    =  5*2    = 5    proved

                                 2             2

         

for k,    5+10 + 15 + ... + 5k =   [5k(k+1)] /2     assumed true

 

k+1   5 + 10 + 15 + ... + 5k  + 5(k+1)  =  [5(k+1) (k+1+1 )]/2   to be proved

 

5 + 10 + 15 + ... + 5k  + 5(k+1)    =

 [5k(k+1) ]/2  + 5(k+1)               =                 

           [5k(k+1)  + 2*5*(k+1)]/ 2 

             5(k+1) [ (k+2)]

                   2    2

 

  1. Prove that n2  2n + 3  for n ≥ 3. Section 2.2 # 27     

n = 3     n2= 9     >= 2(n) + 3 =  9

assume  for  3 <= r <= k     k2 >=  2(k) + 3

prove  (k+1)2 >=  2(k+1) + 3

   (k+1)2 =  k2 + 2k + 1       but k2 > 2k+3

    >= 2k + 3 + 2k + 1   =  4k + 4   LHS

RHS = 2k + 2 + 3  or 2k + 5

subtracting 2k from each side

LHS = 2k + 4         RHS =  5

since k >= 3  2k + 4>= 10   and  10 >= 5  QED

 

  1. Prove that the following statement is true for every positive integer Section 2.2 # 43   

7n-2n is divisible by 5

n = 1   7n-2n   =  71-21  = 7 - 2 = 5

assume   7k-2k  = 5m

prove   7k+1-2k+1 = 5x

7k+1 =  7(7k)  so  7k+1-2k+1 =  7(7k) - -2k+1

but 7k = 5m - 2k    so    7(7k)  -2k+1  =   7(5m - 2k) -2k+1

== 7(5m) - 7(2k) - 2(2k) =   7(5m) - 2k(7 - 2) = 5(7m) - 2k(5)

 

  1. A collection T of numbers is defined recursively by Section 2.4 # 38   

1.       2 belongs to T

2.     if X belongs to T, so does X + 3  and 2*X.

 

Circle the following that belong to T.

 

a.  6             b.  7            c.  19           d.  12

 

  1. The Lucas sequence is defined by         Section 2.4 # 27

L(1) = 1

L(2) = 3

L(n) =  L(n-1) + L (n-2) for n ≥ 3

         

          b.   Write the next three terms of the sequence                                    (3)

               L(3) = 4

               L(4) = 7

c.   Prove that L(n) = F(n + 1)  + F (n -1) for n  ≥ 3

for lower bound

     n = 3    L(3) = F(4) + F(2) =     L(3) = 4    F(4) + F(2) = 3 + 1 = 4

assume  3 <= r <= k     L(k) = F(k+1) + F(k-1)

prove:  L(k+1) = F(k+2) + F(k) 

by definition  L(k) = L(k-1) + L(k-2)

                             L(k+1) = L(k+1-1) + L(k+1-2)

                             L(k+1) = L(k) + L(k-1)

 

 


L(k) = F(k+ 1) + F(k-1)      L(k-1) =  F(k-1 + 1) + F(k-1 -1) = F(k) + F(k-2)

L(k+1) = L(k) + L(k-1)  =  F(k+1) + F(k-1) + F(k) + F(k-2)

                                      =  F(k+1) + F(k) + F(k-1) + F(k-2)

                           =      F(k+2)     +      F(k)

  F(k+2) + F(k) = F(k+2) + F(k)  q.e.d

 

  1. A family has four (4) children; boys and girls are equally likely offspring.  Section 3.5 #43 variant, #47
    1. What is the probability that the oldest child is a boy?                 (5)

 

½

 

b.  What is the probability of exactly three (3) girls?                      (5)

 

bbbb  bbbg   bbgb   bbgg 

bgbb  bgbg   bggb    bggg                    4/16

gbbb  gbbg   gbgb    gbgg

ggbb  ggbg    gggb    gggg

 

  1. Find the indicated term in the expansion (Note: you do not have to simplify) Section 3.6 #2 variant

a.       The third term in (a + b) 7                                       (5)

                    21a5b2

 

b.       The last term in  (ab + 3x) 6                                              (5)

    (3x6)

 

 

 

  1.   In a drug study of a group of patients, 17% responded positively to compound A,, 34% responded positively to compound B, and 8% responded positively to both.   Section 3.5# 42

 

    1. What is the probability that a patient responds positively to compound A given that he or she responds positively to compound B?               (5)

 

P(A and B)   =  8%

                                       P(B)           34%         

 

    1. What is the probability that  a patient responds positively to either compound A or compound B?                                                 (5)

 

P (A or B) =  P(A) + P(B) – P(A and B)   = 17 + 34 – 8 =  43%

 

    1. What is the probability that a patient does not respond positively to either compound?                                                                             (5)

100% - 43% = 57%

 

  1.   Let S = {0,1,2,4,6}  tell whether the following binary relation on S is: 

Section 4.1 # 8b                                                                        (5)

    1. reflexive  a.  not reflexive,   (0,0) not in relation
    2. symmetric     symmetric --  (a,b) and (b,a) in
    3. anti-symmetric    not-antisymmetric --  a /=  b
    4. transitive  not transitive   (2,4) and (4,2) does not imply (2,2)

 

r = { (0,1), (1,0), (2,4), (4,2), (4,6), (6,4)}

 

  1. Let S be the set of people at James Madison University.  Tell whether the following binary relation on S is:  Section 4.1 # 9h
    1. reflexive   not reflexive  - person isn't own brother
    2. symmetric  not symmetric -  Sam is brother of Alice, but Alice isn't brother of Sam
    3. anti-symmetric  not-anti-symmetric
    4. transitive  not transitive  if sam is brother of joe and joe is brother of sam, sam isn't brother of sam.

 

x r y « x is the brother of y

 

  1. Given a function f: S T 

where T = {1,3,5,7,9} and 

where f = { (6,3), (8,1), (0,3), (4,5), (2,7)}

    1. what is the domain of f?    S =  {0,2,4,6,8}
    2. what is the codomain of f? T = {1,3,5,7,9} 
    3. what is the range of f?    {1,3,5,7}
    4. what is the image of 8?   1   
    5. what is/are the preimage(s) of  3?    6,0
    6. is f one-to-one?       no
    7. is f an onto function?    no

 


 

  1.   Construct a PERT chart from the following task table

 

Task

Prerequisite Tasks

Time to Perform

A

E

3

B

C,D

5

C

A

2

D

A

6

E

None

2

F

A,G

4

G

E

4

H

B,F

1

 

  1.   Compute
    1. the minimum time to completion and

    17

 

    1. the nodes on the critical path for your PERT chart.

E A D B  H

  1.   Find a topological sort from your PERT chart.

  E A C D G F B H      or   E A D G  F C B H   or  ...