CS2022/MA2201
HW#9
DUE: Thursday, October 14
1. (2 points) Which of the following are partitions of cfw_1,2,3,.,10
(a) cfw_2,4,6,8, cfw_1,3,5,9, cfw_7,10
(b) cfw_1,2,4,8,cfw_2,5,7,10,cfw_3,6,9
(c) cfw_3,8,10, cfw_1,2,5,9, cfw_4,7,8
(d) cfw_1, cfw_2,,.,cfw
CS2022/MA2201
HW#8 SOLUTIONS
1) (a) C ( n,0) p 0 (1 - p )
n- 0
n
(b)
C ( n, k ) p (1 - p)
n- k
k
= (1 - p )
n
. Since this event is the complement of the event of having
k =1
exactly 0 successes, its probability is 1 - (1 - p ) .
(c) This is the probabil
CS2022/MA2201
HW#8 SOLUTIONS
1. a This is an equivalence relation.
b This is an equivalence relation.
c This is not an equivalence relation since it is not transitive.
d This is not an equivalence relation since it is not transitive.
e This is not an equi
CS2022/MA2201
HW#8
DUE: Friday, October 8
1. (8 points) Do Exercise 4.5.26 from our text.
2. (3 points) Do Exercise 4.5.32 from our text.
3. (14 points) Determine whether each of the following binary relations is:
(1) reflexive, (2) symmetric, (3) antisym
CS2022/MA2201
HW#8
DUE: Monday, April 30
1. (8 points) Adapted from Rosen. Consider the following relations:
R1 = cfw_( x, y ) 2 | x > y
R2 = cfw_( x, y )
R3 = cfw_( x, y )
R4 = cfw_( x, y )
R5 = cfw_( x, y )
R6 = cfw_( x, y )
2
2
2
2
2
| x y
| x < y
CS2022/MA2201
HW#8
DUE: Tuesday, October 14 (Covers graphs.)
1. (8 points) For which values of n are these graphs bipartite?
(a) Kn
(b) Cn
(c) Wn
(d) Qn
(Note that this is Exercise 8.2-24 of our text.)
2. (6 points) For any graph G with v vertices and e e
CS2022/MA2201
HW#7 SOLUTIONS
1) (a) 1/64 (b) 3/64 (c) 1/8.
2) The probabilities of the three outcomes are not equal. The probability that both are
heads is .
3) (a) 10/105 (b) 95/105.
4. (a) 15/40
(b) 35/40
(c) There are 15*14 ways to choose two history b
CS2022/MA2201
HW#8
DUE: Tuesday, October 16
1. (5 points) Do Exercise 6.5.2 (the second Exercise on pg. 413) of our text.
2. (3 points) Do Exercise 6.5.12 of our text.
3. (12 points) A graph is a cubic graph if it is simple and every vertex has degree 3.
CS2022/MA2201
HW#7 SOLUTIONS
1. The CONJECTURE is true. If R and S are reflexive, then
cfw_a, a | a A R and
cfw_a, a | a A S . Therefore, cfw_a, a | a A R S , so R S
2. The CONJECTURE is true. If R is reflexive, then
cfw_a, a | a A R S , so R S
is reflexi
CS2022/MA2201
HW#7
DUE: Monday, October 4
1. (3 points) Suppose you and a friend each choose at random an integer between 1 and 8.
For example, some possibilities are (3,7), (7,3), (4,4), (8,1), where your number is written
first and your friend's number
CS2022/MA2201
HW#7
DUE: Monday, April 23
1. (3 points) Assume that there are 65 students in our class, and I draw 3 people
randomly. Also assume that none of them were born in a leap year. What is the
probability that none of them have the same birthday?
CS2022/MA2201
HW#7
DUE: Thursday, October 9 (Covers relations, Chapter 7.)
1. (8 points) Do Exercise 7.1-4 on pg. 480 of our text.
2. (16 points) Do Exercise 7.1-6 on pg. 480 of our text.
3. (8 points) Assume R and S are each binary relations from set A t
CS2022/MA2201
HW#7
DUE: Thursday, October 11
1. (4 points) Prove or give a counterexample to the following:
CONJECTURE: For any set A and any binary relations R and S on A, if R and S are
reflexive, then so is R S .
2. (4 points) Prove or give a counterex
CS2022/MA2201
HW#6 SOLUTIONS
1. There are 410 possible answer sheets, so 2*410+1 students in the class would assure that
at least three people had the same answer sheet.
2. a Let Ek , 1 k 100 , denote the event that you all pick k. For each k, the probabi
CS2022/MA2201
HW#6 SOLUTIONS
1. If n=0, then there are no strings. If n=1, then there is one string, namely 1. If n 2 ,
then there are n-2 choices for the intermediate bits. Hence, the number of strings is 2n-2.
2. (a) 14
(b) 27
3. 38+23-7=54
4. (a) The t
CS2022/MA2201
HW#6
DUE: Tuesday, September 28
1. (2 points) Do Exercise 4.1.14 from our text.
2. (2 points) You are drawing n cards from a playing deck of 52 cards. What is the
minimum value of n to assure that you draw:
(a) a pair
(b) three of a kind
3.
CS2022/MA2201
HW#6
DUE: Monday, April 16
1. (8 points) Consider the set of all functions f : cfw_1,., n cfw_0,1 where n
+
.
a How many such functions are there?
b How many such one-to-one functions are there?
c How many such onto functions are there?
d H
CS2022/MA2201
HW#6
DUE: Friday, October 3 (Covers material through the end of Chapter 5.)
For these problems, you neednt evaluate binomial coefficients. That is, you may leave terms
like C ( 852,255 ) in your answer.
1. (4 points) What is the probability
CS 2022/ MA 2201 Discrete Mathematics
A term 2014
Solutions for Homework 5
1. Prove the combinatorial identity
n
r
r
n
=
k
k
nk
,
rk
whenever n, r and k are positive integers with r n and k r,
(a) using an algebraic argument (i.e. using the formula for bi
CS2022/MA2201
HW#6
DUE: Friday, October 5
1. (3 points) A professor teaching a Discrete Math course gives a multiple choice quiz
that has 10 questions, each with 4 possible responses: a, b, c, d. What is the minimum
number of students that must be in the
CS2022/MA2201
HW#5 SOLUTIONS
1. (a) P(1) is 1*1!=2!-1=1 which is true.
(b) P(4) is 1*1!+ 2*2!+ 3*3!+ 4*4! = 5!- 1 which is equvalent to 1 + 4 + 18 + 96 = 119
which is true.
n
(c) P(n) is
k * k ! = (n + 1)!- 1
k =1
n+ 1
(d) P(n+1) is
k * k ! = (n + 2)!-
CS2022/MA2201
HW#5 SOLUTIONS
1. a yes
b By the rule of product, we iterate (134 times) the Cartesian Product of the set of all
possible birthdays, yielding 365135=
811963863975903839603110839480062259460906808202672019675805113936
545936783070345584904841
CS2022/MA2201
HW#5
DUE: Tuesday, April 10
1. (6 points) Let ( a1 ,., an ) be a sorted list of numbers. That is, i, 1 i < n, ai ai +1 .
a Give an algorithm with execution time in O (1) to find a number not in the list. That is,
your algorithm should return
CS 2022/ MA 2201 Discrete Mathematics
A term 2014
(Last) Homework 5, due Monday, October 13
READING: Chapters 6, 7, 9 and 10.
1. Prove the combinatorial identity
n
r
r
n
=
k
k
nk
,
rk
whenever n, r and k are positive integers with r n and k r,
(a) using a
CS2022/MA2201
HW#5
DUE: Thursday, September 23 (Professor Ruiz birthday)
s
1. (10 points) Suppose you believe that 1*1!+ 2*2!+ . + n * n ! = ( n + 1)!- 1 for all positive
integers n.
(a) Write P(1). Is P(1) true?
(b) Write P(4). Is P(4) true?
(c) Write P(
CS2022/MA2201
HW#4
DUE: Friday, September 17
1. (4 points) Do Exercise 1.8.4 of our text.
2. (4 points) Do Exercise 1.8.10 of our text.
3. (5 points) Prove or give a counterexample to the following
CONJECTURE: For any f : , g : , if (" n) f ( n) g (n ), t
CS2022/MA2201
HW#5
DUE: Monday, September 29
1. (3 points) Suppose that there is a group of 32 men and n women. Each man knows exactly 5
women, and each woman knows exactly 8 men. What is the value of n?
2. (8 points) Do EXERCISE 4.1.28 on page 311 of our
CS2022/MA2201 HW#5
DUE: Monday, October 1 1. (6 points) Assume that nobody in CS2022/MA2201 is born on a leap year, and there are 135 people in the class. a Is it possible that no two people in this class are born on the same day? b In how many ways can w
CS 2022/ MA 2201 Discrete Mathematics
A term 2014
Solutions for Homework 4
1. Exercise 18 on page 330.
Solution:
a.) Plugging in n = 2, we see that P (2) is the statement 2! < 22 .
b.) Since 2! = 2, this is the true statement 2 < 4.
c.) The inductive hypo
CS2022/MA2201
HW#4 SOLUTIONS
1. For all x 5 , 17 2 x , so 2 x + 17 2 x + 2 x = 2*2 x 2*3x . So choosing C=2 and k=5
yields that 2x+17 is O(3x).
2. Since (" x 1)x 3 x 4 , then x3 is O(x4). If x4 were O(x3), then there exist C, k such
d
that x 4 Cx 3 for al