UMBC CMSC203, Discrete Structures, Fall 2010
Homework Assignments
Notation:
Throughout, we will use these references:
 Rosen 6/e, Rosen 5/e and Rosen 4/e indicate
respectively the sixth, fifth and fourth editions of
Discrete Mathematics and Its Applications, Kenneth Rosen,
McGrawHill.
 Epp 3/e and Epp 2/e indicate respectively the
third and second editions of
Discrete Mathematics with Applications, Susanna S. Epp, Thomson
Publishing.
 [Adapted from Rosen 6/e]
State whether each of the following relationships between
sets is true or false. Justify your answer briefly.
(Here, ∅ is the empty set.)
 ∅ ∈ ∅
 {∅} ∈ {∅}
 ∅ ∈ {∅}
 {∅} ⊆ {∅}
 ∅ ⊆ ∅
 [Adapted from Epp 3/e]
Let R be the set of all real numbers and let
A indicate
the complement of the set A. We define the sets A,
B and C as follows:
A = { x ∈ R  3 ≤ x ≤ 0 }
B = { x ∈ R  1 < x < 2 }
C = { x ∈ R  6 < x ≤ 8 }
Describe the following sets:
 A ∪ B
 A ∩ C

A
∪
B
 A ∩
B

A ∪ B
 For each of the following functions, state whether the function
is onetoone, whether the function is onto and whether the function
is a bijection. Pay close attention to the domain and codomain
of each function. Briefly justify your answer.
 f : N → N,
f (n) = n^{2} + 5.
 f : Z → Z,
f (n) = n^{2} + 5.
 f : N → N,
f (n) = 2 n + 7.
 f : R → R,
f (x) = 2 x + 7.
 f : R → R,
f (x) = 2 x^{3} + 1.
Note: N, Z and R denote the set of natural numbers,
integers and real numbers respectively.
 Read "A
Mathematician's Lament" (PDF) by Paul Lockhart posted on the web
site of the Mathematical Association of
America. (Here's a local copy.)
In this essay, Lockhart complains that in the standard
mathematics curriculum taught in elementary school through high school
"there is no actual mathematics being done in our mathematics classes"
(p. 14) and that "there is surely no more reliable way to kill
enthusiasm and interest in a subject than to make it a mandatory part
of the school curriculum." In short, he claims that the mathematics
education you have likely received so far is all wrong.
Defend yourself! Write a convincing argument against Lockhart's
position. Be persuasive. Pick apart Lockhart's argument. Support
your own position with facts, personal experience, logical
reasoning, etc.
Please submit this part of your homework on its own pieces of paper
as it will be graded separately from other questions.
 Vertex Cover.
Consider a graph G. Let X be a subset of the vertices
in G. We say that X is a vertex cover of
G if for every edge in G at least one of the endpoints
of that edge is in X.
In the graph below, find a vertex cover with as few vertices as you can.
List the vertices of the vertex cover you found and briefly argue that it
is the smallest possible.
Hint: the smallest vertex cover in this graph has 10 vertices.
 Hamiltonian Circuits.
[Adapted from Problem Solving Through Recreational Mathematics by
Averbach & Chein, 1980.]
A graph has a Hamiltonian circuit if there is a path in the
graph that visits every vertex exactly once and returns to the first vertex
in the path.
 Does the graph below have a Hamiltonian circuit? Justify your answer.
 In general, if a graph has a Hamiltonian circuit, is it necessarily the
case that the graph has an Euler circuit? Justify your answer.
 In general, if a graph has an Euler circuit, is it necessarily the
case that the graph has a Hamiltonian circuit? Justify your answer.
 Regular Graphs.
In a dregular graph, every vertex in the graph has degree d.
Recall that the degree of a vertex is the number of edges incident on the
vertex. (I.e., count the number of edges coming out of a vertex and that is
its degree.)
 Draw a 3regular graph with 6 vertices.
 Draw a 3regular graph with 8 vertices.
 Draw a 3regular graph with 10 vertices.
 Are there any 3regular graphs with 9 vertices?
why or why not?
 Tautologies. Use a truth table to show that the following
proposition is a tautology. You must show the intermediate steps in
the truth table.
(¬ (p ∨ q)) ⊕
( (¬p) → q)
 Tautologies, again.
[From Rosen 5/e.]
Show that the following proposition is a tautology using algebraic
manipulation of logical equivalences (i.e., without using a
truth table).
You may use the logical equivalences in Tables 5, 6 and 7 in
Section 1.2 of Rosen 5/e or Tables 6, 7 and 8 in Rosen 6/e.
Show all your work.
[
( p ∨ q)
∧
( p → r)
∧
( q → r)
]
→ r
 Logical Equivalences.
[From Rosen 5/e.]
Show that ¬ p → (q → r)
and q → (p ∨ r) are logically
equivalent without using truth tables.
You may use the logical equivalences in Tables 5, 6 and 7 in
Section 1.2 of Rosen 5/e or Tables 6, 7 and 8 in Rosen 6/e.
 Knights and Knaves.
[From "A Whole Slew of ComputerGenerated Knights and Knaves Puzzles"
by Zac Ernst, 1999.]
Statements made by knights are true. Statements made by knaves are
false. You meet three people: Xavier, Yolanda and Zain. You know that
each is either a knight or a knave. This is what they said:
Xavier: "It is not the case that Zain is a knave."
Yolanda: "Zain and Xavier are both knights."
Zain: "Xavier is a knight or Yolanda is a knave."
Which of Xavier, Yolanda and Zain are knights? which are knaves?
Show your reasoning.
For this homework assignment, you are asked to provide 4 proofs.
Remember that proofs are written in English. You proof should
not be a sequence of arithmetic equations. There must be a narrative
composed of complete English sentences, correctly punctuated, with
math symbols mixed in as appropriate, which convinces the reader
that the claim is correct.
 Indirect Proof.
Give an indirect proof for the following claim:
If m and n are odd integers, then m⋅n
is an odd integer.
 Proof by Contradiction.
Prove by contradiction that the following graph is not 3colorable.
 Proof by Cases. [Adapted from Rosen 5/e.]
Let min: R × R → R be the function that
"returns" the minimum of two values.
(Here, R is the set of real numbers.)
For example,
min(3.1, 5) = 3.1
min(17.2, 9.4) = 9.4
Prove by cases, that for all real numbers a, b and c,
that
min(min(a, b), c) =
min(a, min(b, c))
 Equivalence Proof. [Adapted from Rosen 5/e.]
Prove the set equality below by showing that every element of
the set on the left hand side of the equality is also an element of the set
on the right hand side, and vice versa. (I.e., do not prove this using
algebraic identities.)
(B − A) ∪ (C − A)
=
(B ∪ C) − A
 Indirect Proof 2.
For some function
f : R → R,
let
g : R → R
be defined by g(x) = f (x/2).
(Note: here R is the set of real numbers.)
Provide a wellwritten indirect proof which shows that if f
is onetoone then g must also be onetoone. Your proof
must follow the indirect proof format.
 Arithmetic with mod. Use the repeated squaring technique
to compute 96^{19} % 149. Show all of your work. Your work
should not have any numbers bigger than 149^{2} = 22201.
 Inverses mod 29. For each integer x,
1 ≤ x < 29, find an integer y,
1 ≤ y < 29, such that
x ⋅ y ≡ 1 (mod 29).
In other words, x ⋅ y % 29 = 1.
Then, x and y are called inverses modulo 29.
 A Shortcut. Fermat's Little Theorem states that
if p is prime, then for any integer a not divisible
by p,
a^{ p − 1} ≡ 1 (mod
p).
Take advantage of Fermat's Little Theorem to compute the value
9^{5282} % 19 by noticing that 9^{18} ≡ 1 (mod
19).
Show your work.
 GCD Proof.
Let a and b be integers such that a is even
and b is odd. Argue that gcd(a, b) =
gcd(a/2, b).
 Euclid's Algorithm.
Use Euclid's algorithm to compute gcd(18893511, 1154300). Show all of your work.
 Extended Euclid's Algorithm.
Use the Extended Euclid's Algorithm to find the multiplicative inverse of
173 modulo 235012. Show all of your work.
(See Notes on the Extended Euclid's
Algorithm.)
 Chinese Remainder Theorem.
Use the Chinese Remainder Theorem to find an integer x such that
0 ≤ x < 1456 and the following congruences hold:
x ≡ 5 (mod 7)
x ≡ 8 (mod 13)
x ≡ 11 (mod 16)
Note that 7 ⋅ 13 ⋅ 16 = 1456 and recall that the notation
a ≡ b (mod n)
means that a % n = b % n, where
% is the remainder operator. Show your work.
(See notes on the Chinese Remainder Theorem.)
For better typesetting, you can download this homework set in PDF:
hw7.pdf.
Instructions:
In the following questions you are asked to use proof by induction.
Your proof must not simply be a sequence of equations, even if the
statement you are proving is arithmetic in nature. Clearly indicate
using complete English sentences:
1) what you are allowed to assume from the induction hypothesis,
2) what you need to show to establish the induction step, and 3)
which steps of the proof uses the induction hypothesis.
Responses that do not include wellwritten English sentences that clearly
explain your proof will receive a grade of no more than 50%.
 Induction (cubes).
Prove by induction that for all integers n ≥ 1
1^{3} +
2^{3} +
3^{3} +
⋅⋅⋅
n^{3} = [ n (n+1) / 2]^{2}
 Induction (fractions).
Prove by induction that for all integers n ≥ 1
 Regular Graphs, Revisited.
Recall that in graph theory, a kregular graph is an undirected
graph where every vertex has degree k. Here we do not allow
edges from a vertex to itself (self loops) and we do not allow more
than one edge between a pair of vertices. Also recall that the
degree of a vertex is simply the number of edges incident on that
vertex.
Prove by induction on k that for every k ≥ 0, there exists
a kregular graph.
 Induction (inequality).
Let x ≥ 0 be a real number.
Prove by induction on n, that for all n ≥ 2,
1 + nx ≤ (1 + x)^{n}.
Hint: Do not expand (1 + x)^{n} all the way.
Use induction!
Note: the inequality actually holds for x ≥ −1, but
for this exercise you will only prove the easy case.
Instructions:
In the following questions you are asked to use proof by induction.
Your proof must not simply be a sequence of equations, even if the
statement you are proving is arithmetic in nature. Clearly indicate
using complete English sentences:
1) what you are allowed to assume from the induction hypothesis,
2) what you need to show to establish the induction step, and 3)
which steps of the proof uses the induction hypothesis.
Responses that do not include wellwritten English sentences that clearly
explain your proof will receive a grade of no more than 50%.
 Fibonacci Numbers. [Adapted from Rosen 5/e]
The Fibonacci numbers
f_{0}, f_{1},
f_{2}, ...
are defined recursively by
f_{0} = 0,
f_{1} = 1,
f_{2} = 1 and for all n ≥ 3,
f_{n} = f_{n − 1} +
f_{n − 2 }.
Use structural induction to show that
( f_{1} )^{2} +
( f_{2} )^{2} +
⋅⋅⋅ +
( f_{n} )^{2}
=
f_{n}
f_{n+1}
 String Reversal. [Adapted from Rosen 5/e]
In this problem, we consider only strings over {0, 1} of length
greater than or equal to 0. For example, the string 01001 is a
string of length 5. We use λ to denote the empty string.
For a string w, we can define the reversal of the string,
written w^{R}, as follows:
 λ^{R} = λ
 for a string x, ( 0 x )^{R} =
x^{R} 0.
 for a string x, ( 1 x )^{R} =
x^{R} 1.
For example, we can unwind the recursive definition of 01001^{R}
by:
01001^{R}
= 1001^{R}0
= 001^{R}10
= 01^{R}010
= 1^{R}0010
= λ^{R}10010
= λ10010
= 10010.
Use structural induction to prove that for all strings
w and x, (w x)^{R}
= x^{R} w^{R}.
Note:
For this problem, you may assume without proof that
string concatenation is associative. So, given 3 strings
x, y and z,
( x y ) z
=
x ( y z ).
 Tournament Graphs. [Adapted from Rosen 5/e]
Recall that a tournament graph G is a directed
graph (without self loops) such that for any two distinct vertices
u and v in G, exactly one of (u,
v) or (v, u) is an edge in G.
(Here, u and v being distinct means
u ≠ v.)
You can think of the vertices of a tournament graph as teams
participating in a roundrobin tournament (where each team plays
every other team once) in a sport that does not allow tie games.
Then, for any two teams u and v, either u
defeats v or v defeats u (but not both).
If u defeats v, then the edge (u, v)
is placed in the graph. Otherwise, v defeats u and
(v, u) is placed in the graph.
In this problem, you are asked to prove by induction that every
tournament graph G has a "winner". Here, a winner
is a vertex w that defeats every vertex (other than
w) either directly or indirectly. We say that w
defeats x directly if (w, x) is an
edge in G. Vertex w defeats vertex x
indirectly if there exists a vertex u such that
(w, u) and (u, x) are edges in
G (that is, w defeats u which in turn defeats
x).
Note: a tournament graph can have more than one winner.
Hint: for a vertex v think about the set of vertices
that v defeats directly and a second set of vertices that
v defeats indirectly but not directly.
For each of these questions, you must show your work and explain
your answer. Answers that consist of a single number will receive
fewer than 50% credit.
When factorials are involved, leave your answer in terms of factorials
(e.g., 5!/(3! ⋅ 2)).
 Trip Planning. For a trip, you packed 2 pairs of
sneakers, 3 pairs of jeans, 4 pairs of shorts, 5 shirts, 2 sweaters
and 3 rings. How many different outfits can you make from the
clothing you packed, if an "outfit" consists of 1 pair of sneakers,
1 pair of jeans or shorts (but not both), 1 shirt, at most 1 sweater
and any number of rings? (Note: wearing rings on different fingers
does not count as a different "outfit.")
Note: Assume that your sneakers, jeans, shorts, shirts,
sweaters and rings are distinguishable.
 Sock Drawer.
Suppose your sock drawer has 17 pairs of socks that are black, white
or tan. Which of the following statements must be true? Justify
your answer.
 There are at least 5 pairs of black socks, at least 5
pairs of white socks and at least 5 pairs of tan socks.
 There are at most 4 pairs of black socks, at most 4 pairs
of white socks or at most 4 pairs of tan socks.
 There are at least 6 pairs of black socks, at
least 6 pairs of white socks or at least 6 pairs of tan
socks.
 Another Sock Drawer.
Suppose that your sock drawer has 6 pairs of black socks, 5 pairs of
white socks and 6 pairs of tan socks. How many different ways are there to
pack 5 pairs of socks? You can bring as many pairs of socks of each color
as you want. Assume that socks of the same color are not
distinguishable.
 Hotel Room Closet.
In the hotel room, you hang up in the closet the 5 shirts and 2
sweaters that you packed. How many ways can you arrange the shirts
and sweaters in the closet (from left to right) so that the 2
sweaters are adjacent to each other? (As before, assume that your
shirts and sweaters are distinguishable.)
For each of these questions, you must show your work and explain
your answer. Answers that consist of a single number will receive
fewer than 50% credit.
When factorials are involved, leave your answer in terms of factorials
(e.g., 5!/(3! ⋅ 2)).
 Dog Sled Team.
We wish to form a sled dog team with 7 dogs chosen from a group of
14 Siberian Huskies and 9 Alaskan Malamutes. How many different
teams can be formed if we must have more Siberian Huskies than
Alaskan Malamutes?
Note: in this question the dogs are
considered individuals and are distinguishable.
 Car Park.
There are 4 red cars and 7 blue cars parked in 11 consecutive spots
of a parking lot. Each car is distinguishable by its license plate.

How many ways can the cars be parked in these 11 spots so the 4
red cars are adjacent to each other?

How many ways can the cars be parked in these 11 spots so none
of the 4 red cars is adjacent to another red car?
 Marble Placement.
In a board game, you have 19 indistinguishable marbles that you can
place in 5 distinguishable locations. You must place at least 2
marbles at each location, but are otherwise allowed to place as
many or as few marbles at each location. How many different ways
can you make these placements?
 Balls & Bins.
You have 13 balls that you throw at 5 bins labeled A, B,
C, D and E. Our assumption is that when a ball is thrown at the
bins, there is an equal probability that the ball lands in any
particular bin. Also, the ball will always land in one of the
bins. Each bin is large enough to hold any number of balls.

You throw the 13 balls, one at a time, at the bins. What is the
probability that exactly 3 balls land in bin A? Justify your
answer.

You throw the 13 balls, one at a time, at the bins. What is the
probability that 4 or fewer balls land in bin B? Justify your
answer.
For each of these questions, you must show your work and explain
your answer. Answers that consist of a single number will receive
fewer than 50% credit.
 Two Urns. [Adapted from Epp, 3/e.]
You have two urns. One urn holds 5 red balls and
13 yellow balls. The second urn holds 9 red balls and
11 yellow balls. You pick one ball using this procedure:
randomly pick one of the two urns with equal probability, then
pick a ball from the chosen urn so that each ball is chosen
with equal probability.
 What is the probability that the chosen ball
is red?
 If the chosen ball is red, what is the
probability that the chosen ball came from the first urn?
 Independent Dice.
A standard 6sided die has faces numbered 1, 2, 3, 4, 5 and 6.
Assume that it is equally likely for each of the faces to be on top
when the die is rolled. Suppose that you roll 2 six sided dice and
add the two numbers that appear on top. Let E be the event
that the sum of the two dice is divisible by 3. Let F be the
event that the sum is less than or equal to 6. Are events E
and F independent?
 Odd Man Out.
Four friends play a game called Odd Man Out. They each flip a fair
coin. If 1 person has heads and the other 3 have tails, then the
person with heads is the odd man. Similarly, if 1 person has tails
and the other 3 have heads, then the person with tails is the odd
man. What is the probability of having an odd man after each person
flips just once? Explain your answer.
 Odd Man Out, Part Deux.
Suppose the four friends in the question above continue playing
Odd Man Out until someone becomes the odd man. What is the expected
number of times that they have to play this game? Explain
your answer.
Note: this assignment is due on
Thursday after Thanksgiving (not Tuesday).
 4 and 8sided Dice.
You have a fair 4sided die and a fair 8sided die.
(Here a die is fair if when the die is rolled, there
is an equal probability for any particular side to be on top.)
The sides of the 4sided die are labeled with 2, 7, 9 and 12.
The sides of the 8sided die are labeled with 1, 2, 3, 4, 7, 8, 9 and 10.
Note: Show all of your work and explain your answers.
 What is the expected value of the number on top
when you roll the 4sided die?
 What is the expected value of the number on top
when you roll the 8sided die?
 What is the expected value of the sum
of the numbers on top when you roll both dice?
 Plastic Utensils.
You randomly pick utensils from a box with plastic knives, forks
and spoons. Each time you pick, there is an equal probability of
picking any of the utensils remaining in the box. Initially, the box
holds 4 forks, 3 spoons and 7 knives.
Note: Show all of your work and explain your answers.
 Suppose you pick 3 utensils without replacement.
What is the probability that you picked a fork, a spoon and
a knife (in any order)?
 Suppose that you pick 2 utensils without replacement.
What is the
conditional probability
that the second utensil you picked is a knife?
 Suppose that you pick 2 utensils without replacement.
What is the conditional probability that the second utensil
you picked is a knife given that the first utensil is a fork?
 Suppose that you pick 2 utensils without replacement.
What is the probability that at least one of the two is a spoon?
 Relations and Properties.
For each of the following relations, state whether
relation is reflexive, symmetric, antisymmetric and transitive.
Briefly justify your response in each case.
Note: a relation R on A
is antisymmetric if for all (a, b) ∈ A
× A, (a, b) ∈ R and
(b, a) ∈ R implies that a =
b.

R_{1} = { (a, b) 
a and b are integers and a = 3b }

R_{2} = { (a, b) 
a and b are integers and  a − b 
is even }

R_{3} = { (a, b) 
a and b are integers and a b = 0 }
 Combining Equivalence Relations.
[From Rosen 5/e.]
Suppose that R_{1} and R_{2}
are both equivalence relations on A.
 Must R_{1} ∩ R_{2}
be an equivalence relation? Give a proof or a counterexample.
 Must R_{1} ⊕ R_{2}
be an equivalence relation? Give a proof or a counterexample.
Note: For two sets A and B, A ⊕
B is the symmetric difference of A and B.
That is,
A ⊕ B
=
(A − B)
∪
(B − A) .
 Partial Orders.
Consider the set A = { 2, 5, 6, 8, 10, 24, 30, 120 } and the relation
R on A defined by:
R = { (a, b)  a divides b }.
Since R is reflexive, antisymmetric and transitive, it
is a partial order.
 Draw a Hasse diagram for R.
 List two incomparable elements in R
 What are the minimal and maximal elements of this
partial order?
 Does R have a greatest element? Why or why
not?
 Does R have a least element? Why or why
Last Modified:
2 Dec 2010 02:17:53 EST
by
Richard Chang
to Fall 2010 CMSC 203 Homepage