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D E PA R T M E N T O F C O M P U T E R S C I E N C E S
COLLEGE OF ENGINEERING
FLORIDA TECH
DISCRETE
M AT H E M AT I C S
HOMEWORK
WILLIAM SHOAFF
FA L L 2 0 1 3
Problems Sets
5
Overview
7
Logic
Sets
11
Sequences
13
Recursion
15
19
Summations
Induction
21
Relations
23
Functions
25
Numbering & Naming Systems
Combinatorics & Number Theory
Proofs
Index
35
39
27
31
Overview
Below is a collection of problems that relate to topics in discrete mathematics.
I encourage you to solve some of them. If you find, after a sincere effort, that
you need hints to find a solution or that you simply need the answer to be
explained to you, then use the time honored method: Ask an oracle, perhaps
Pythia, (a nondeterministic Turing machines)1 to provides the answer, and
you, the validator (a deterministic Turing machine)2 checks it is correct.
You can choose to earn extra credit by solving these problems, but there
are rules.
1. You must start earning extra credit before Monday, October 15. In particular, you cannot seek extra credit toward the end of the semester once
you’ve realized you will almost certainly not earn the grade you seek.
2. You must present a nicely written solution to the problem. For extra, extra
credit learn to write mathematics in a document processing system such as
LATEX or (sigh) Word. A system that can nicely typeset mathematical, and
other, notations.
3. You must explain your solution to the instructor’s satisfaction in a solo
interview. Schedule an interview through the department’s administrative
assistant: Mrs. Karen Brown.
Students seek extra credit for many reasons:
• To improve their grade
• To learn advanced topics
• For future recommendation from their instructor
• And there may be other reasons of which I have not thought.
There are many types of students: To many to enumerate, but let me address
the major classes
• To those who want to learn advanced topics, demonstrate your understanding as basic topics by solving these homework problems. You will earn an
A in the course.
1
I am not an oracle. I cannot always give
the right answer to your questions. Some
questions have no answers. I have not
mastered the answers to many questions that
do have answers.
2
I don’t mean to imply anyone is deterministic, I don’t believe that. I am just
trying to describe computational ideas in
anthropomorphic terms.
6 department of computer sciences
college of engineering
florida tech
• To those who want to improve their grade, You can advance at most one
letter grade by solving these extra credit problems. In fact, you can’t
advance from low X to Y, unless your extra credit work is impressive and
you own it: Don’t fall behind.
• To those who want future recommedations: Talk to your instructor frequently
Letter Grade Calculation
The letter grade you receive in this course is based on the work you demonstrate in class.
Your total score can be calculated by the formula
S =
n−1
X
wk sk
k =0
where the non-negative weights wk sum to 1 and the values sk are scores on
your efforts, k = 0, 1, . . . , n − 1. There are ten homework assignments and
one final examination; therefore n = 11. These are summarized below. Each
assignment will have a maximum score of 100. Therefore your score will be
some number from 0 to 100.
I expect to use the range of scores below to determine final letter grades.
A = 90 – 100
B = 80 – 89
C = 70 – 79
D = 60 – 69
F = 0 – 59
However, I will compute some statistics to help me understand the achievement of the class. I will use:
• The mean (average) score µ over all students in the class
µ=
m−1
1 X
Sj
m j=0
where there are m students and student j scored S j points.
• The standard deviation σ of all scores
s
Pm−1
j=0 (S j
σ=
− µ)2
m
• Your z score
S −µ
σ
which measures how many standard deviations your score S is above of
below the mean µ.
z=
• Your C + score
C + = 10z + 79
which assumes an average student will be at the C+/B- cutoff.
• The median m that partitions the scores into two equal-sized groups:
Those below m and those above m.
Logic
1. It is interesting to think of False =0 and True =1 as numerical values
rather than Boolean values. When you do, addition can implemented by
the rules
0 + 0 = 0,
0 + 1 = 1,
1 + 0 = 1,
1 + 1 = 10.
Full addition requires carry-in and carry-out bits. In general, you must be
able to add two terms (bits) a and b and a carry-in bit cin to compute a
sum bit s and a a carry-out bit cout .
(a) Complete the truth table below that describes a full adder: The
Boolean function that adds two bits a, b, and a carry-in bit cin to
produce a sum bit s and a carry-out bit cout .
Input
Output
a
b
cin
s
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
cout
(b) Find a Boolean expression that represents the sum function.
(c) Find a Boolean expression that represents the carry-out function.
(d) Explain how you would use Boolean operations to add two 8-bit
strings a = a7 a6 · · · a1 a0 and b = b7 b6 · · · b1 b0 together with a
sequence of carries c = c7 c6 · · · c1 c0 to product a sum s = s7 s6 · · · s1 s0
and carry out bit c8 .
2. In general, a Boolean function maps an n-tuple of Boolean values to an
m-tuple of Boolean values, as in problem 1 where 3 values are mapped to
2. How many different n-input, m-output Boolean functions are there?
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discrete mathematics homework
3. A quasi-Boolean function maps an n-tuple ( p0 , p1 , . . . , pn−1 ) of Boolean
values to one of three values: 0 = (False), 0.5 = (Maybe) or 1 =
(True).
(a) How many different n-variable quasi-Boolean functions are there?
(b) If you allow quasi-Boolean (0, 1/2, 1) input as well, How many
different n variable functions are there?
4. Use the laws of Boolean algebra: commutative, associative, distributive,
De Morgan’s, etc., and the shorthand p → q for ¬p ∨ q, to prove the
following equivalences.
(a) p → ( q → r ) ≡ ( p ∧ q ) → r
(b) ( p ∧ q ) → r ≡ ( p → r ) ∨ ( q → r )
5. I once wrote some code that looked like this:
bool f(bool p, bool q, bool r) {
if (p) {
if (q) {
return r;
}
return true;
}
return true;
}
One student suggested this refactoring:
bool f(bool p, bool q, bool r) {
if (p && q) {
return r;
}
return true
}
Another suggested the code should be written as:
bool f(bool p, bool q, bool r) {
if (p) {
return r;
}
if (q) {
return r;
}
return true;
Are the code fragments functionally equivalent? Which do you think is the
more beautiful code?
logic
6. Let X and Y be sets with cardinalities n and m, respectively. Let P( x, y)
be a predicate meaning x is related to y. There are eight ways to quantify
P( x, y) using (∀x) (∀y), (∃x), and (∃y). These are shown in the tree
diagram below.
(∀x)
(∃x)
(∃x)
(∀x)
(∀y)
(∃y)
Quantification
(∀x)
(∃x)
(∀y)
(∃y)
(∃y)
(∀y)
(a) Draw graph diagrams for each of the quantified phrases.
i. (∀x)(∀y)( P( x, y))
v. (∀y)(∀x)( P( x, y))
ii. (∀x)(∃y)( P( x, y))
vi. (∀y)(∃x)( P( x, y))
iii. (∃x)(∀y)( P( x, y))
vii. (∃y)(∀x)( P( x, y))
iv. (∃x)(∃y)( P( x, y))
viii. (∃y)(∃x)( P( x, y))
(b) Use your graphs to argue that
i. (∀x)(∀y)( P( x, y)) ≡ (∀y)(∀x)( P( x, y))
ii. (∃x)(∃y)( P( x, y)) ≡ (∃y)(∃x)( P( x, y))
iii. None of the other four quantifications are equivalent.
(c) You know the rules of Suduko, or if you don’t look them up. Pretend you want to write a program that plays the game. Devise simple
predicate statements that allow you to write the rules of the game.
Use predicates such as Square ( s) to say “s is a square on the board”;
and SameRow ( s0 , s1 ) to say “s0 and s1 are in the same row”. Let
board ( s) denote the value at square s.
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Sets
1. Pretend you are writing traffic accident software and want to categorize
accidents by the day of the week on which they occur. Pretend there are n
accident reports to categorize.
(a) What is the size of the sample space? That is, in how many ways can
the n accident reports be distributed over 7 days?
(b) In how many ways can all n accidents occur on one single day?
(c) In how many ways can all n accidents occur on only two days?
(d) Let’s looks at the other end: In how many ways can all n accidents
occur on seven, and no less, days.
(e) Show that for n ≥ 0
Note that
k!
7
X
k =0
k!
( )
!( ) X
7
n
7 n
= 7n
=
7k
k
k k
k =0
where 7k = 7 · 6 · · · (7 − k + 1).
2. There are 2000 students on campus who own Team Fortress 2, Plants vs
Zombies, or Kerbal Space Program. If 500 students owe all three games,
200 own only Team Fortress 2, 350 own only Plants vs. Zombies, and 150
own only Kerbal Space Program, how many of these games in total are
owned by Florida Tech students?
3. The axiom of choice states that for any collection X of nonempty sets,
then there exists a choice function c() that selects an element of A for
every set A ∈ X.
(a) Give an example of a choice function on the collection
X = {{0, 1} , {2, 3} , {4, 5}}
(b) How many choice functions could you define on the set X in problems 3a?
(c) Let Z = {nk : k ∈ Z}. Let X be the collection
X = {Z : n ∈ N}
Give an example of a choice function on X.
!
7
7
= k!
k
k!(7 − k)!
= 7 · 6 · · · (7 − k + 1)
= 7k
is called 7 falling 2. In general, nk is called
falling factorial: n falling k.
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discrete mathematics homework
(d) Write the axiom of choice using predicate logic notation.
(e) The Cantor C set is defined recursively by removing the middle third
of a set of intervals.
C0 = [0, 1]
1
2
C1 = [0, ] ∪ [ , 1]
3
3
1
2 1
2 7
8
C2 = [0, ] ∪ [ , ] ∪ [ , ] ∪ [ , 1]
9
9 3
3 9
9
Cn−1 2 + Cn−1
Cn =
∪
3
3
C0 2 + C0
∪
3
3
C1 2 + C1
=
∪
3
3
=
i. What is a choice function for Cn ?
ii. What is the total length (measure) of the intervals in Cn ?
(f) The Cantor set C is
C = lim Cn
n→∞
i. What is the total length (measure) of C?
Sequences
~ has terms pn defined by a polynomial
1. A polynomial sequence P
pn = p ( n ) =
m−1
X
ak nk
k =0
(a) Let p(n) = an + b be a linear polynomial, and let
hl0 , l1 , l2 , . . .i = hb, a + b, 2a + b, . . .i
be an arithmetic sequence. Show that ln − ln−1 = a for n ≥ 1.
(b) Let p(n) = an2 + bn + c be a quadratic polynomial, and let
hq0 , q1 , q2 , q3 . . .i = hc, a + b + c, 4a + 2b + c, 9a + 3b + c, . . .i
be a quadratic sequence. Show that qn+1 − 2qn + qn−1 = 2a for n ≥ 1.
(c) Let p(n) = an3 + bn2 + cn + d be a cubic polynomial, and let
hc0 , c1 , c2 , . . .i = hd, a + b + c + d, 8a + 4b + 2c + d, 27a + 9b + 3c + d, . . .i
be a cubic sequence. Show that cn+2 − 3cn+1 + 3cn − cn−1 = 6a for
n ≥ 1.
(d) Generalize the pattern you have discovered.
2. You have just graduated from Florida Tech with a BS in computer science
or software engineering. Two companies have offered you a job. 3
• Company ABC offers you $50,000 plus a yearly raise of 5% of your
salary.
• Company XYZ offers you $60,000 plus a yearly raise of $3,000.
(a) All other benefits being equal and assuming long term employment,
which company’s offer would you accept?
(b) Does your salary from ABC ever exceed your salary from XYZ?
(c) How long would you need to stay at ABC until your total earnings
equaled your total earnings from XYZ?
3
Recent surveys show most students
graduating from Florida Tech’s Computer
Science Department earn starting salaries
from $50,000 to $70,000.
14
discrete mathematics homework
(d) But there is another problem. A dollar today may not be equal to a
dollar tomorrow. Assume there is a constant 3% rate of inflation. The
value of n dollars today has a future value of only n/1.03 in one year.
Using this model on the value of money, how long must you work at
ABC?
3. Consider the sequence of sequences.
hh0i , h0, 1i , h0, 1, 3, 2i , h0, 1, 3, 2, 6, 7, 5, 4i , h0, 1, 3, 2, 6, 7, 5, 4, 12, 13, 15, 14, 10, 11, 9, 8i , . . .i
(a) How are the sequences generated?
(b) Describe the property of the sequences.
Recursion
1. A partition of positive integer n is a collection of positive integers whose
sum is n. For instance, 6 = 1 + 5 = 2 + 4 = 3 + 3 are three different partitions of 6 into 2 terms. The order of terms in the sum is not considered
relevant.
The terms in a partition are called parts. Let p(n, k) denote the number of
partitions of n into exactly k parts. For example, 6 can be partitioned into
k = 1, 2, 3, 4, 5 and 6 parts in the following ways.
p(6, 1) = 1
(6)
p(6, 2) = 3
(5 + 1, 4 + 2, 3 + 3)
3 ways
p(6, 3) = 3
(4 + 1 + 1, 3 + 2 + 1, 2 + 2 + 2)
3 ways
p(6, 4) = 2
(3 + 1 + 1 + 1, 2 + 2 + 1 + 1)
2 ways
p(6, 5) = 1
(2 + 1 + 1 + 1 + 1)
1 way
p(6, 6) = 1
(1 + 1 + 1 + 1 + 1 + 1)
1 way
1 way
That boundary conditions p(n, 0) = 0, p(n, 1) = 1 and p(n, n) =
1, p(n, n + 1) = 0 state you cannot write n as the sum of no parts or n + 1
parts; and you can write n using 1 or n parts in only one way.
(a) Find a three-term recursion formula the computes p(n, k) as
p(n, k) = p( x, y) + p(z, w)
where x and z are smaller than n, and y and w are no greater than k. Be
able to explain why your recursion is sensible.
(b) Use your recursion formula to fill in the row i = 0 to row i = 9 and
16
discrete mathematics homework
column j = 0 through j = i + 1 of the integer partition triangle.
Partition numbers p(n, k)
k
0 1 2 3 4 5 6 7 8
n
0
1
2
3
4
5
6
7
8
9
9
0
0
0
0
0
0
0
0
0
0
2. A permutation is a one-to-one function from a finite set onto itself. Let
~π = hp0 , p1 , . . . , pn−1 i be a permutation of 0 to n − 1.
D
E
An ascent in a permutation is a pair p j−1 , p j where p j−1 < p j . The pair
is a descent if p j−1 > p j .
For instance,
• h0, 1, 2i has 2 ascents,
• h2, 1, 0i has no ascents,
• Each of h1, 0, 2i, h0, 2, 1i, h2, 0, 1i, and h1, 2, 0i has one ascent.
DE
Let nk be the number of permutations of n values with k ascents. From
the above example, you know
* +
* +
* +
3
3
3
= 1,
= 4,
=1
0
1
2
In general, there is only one permutation of n values with no ascents
* +
n
=1
hn − 1, n − 2, . . . , 0i so
0
And, there is only one permutation of n values with n − 1 ascents
*
+
n
=1
h0, 1, . . . , n − 1i so
n−1
3. Explain why
n−1 * +
X
n
k =0
k
= n!
(1)
DnE
k
k.”
is the Eulerian number called “n ascent
recursion
4. Explain why
* + *
+
n
n
=
.
k
n−k−1
(2)
5. Let ~π be a permutation of n values: 0 to n − 1. Develop a notation for
the permutation that results from inserting a new value n into ~π and some
position j. In how many ways can you insert a new value n into ~π?
6. Let ~π have k ascents. Show there are k + 1 sequences among ~πn, j , j =
0, 1, . . . , n − 1 that have k ascents.
7. Let ~π have k − 1 ascents. Show there are n − k sequences among ~πn, j ,
j = 0, 1, . . . , n − 1 that have k ascents.
8. Use your answers from the previous problems to explain the three-term
recurrence equation
+
+
* +
*
*
n
n−1
n−1
= (k + 1)
+ (n − k )
k
k
k−1
9. Let n be a natural number greater than 0. Consider the sequence generated
by the recurrence equation
!
1
n
xk+1 =
xk +
, k ≥ 0, x0 = 1.
2
xk
(a) Compute terms x1 , x2 , x3 , and x4 , for n = 1, 2, 3, 4.
(b) What values does the recursion compute?
17
Summations
1. The area under a curve can be approximated by the trapezoid rule. 4
For instance the area under the sine curve from x = 0 to x = π can be
approximated by 4 trapezoids with base h = π/4 and heights sin kπ/4,
k = 0, 1, 2, 3, 4.
4
How can you compute the area of a
trapezoid?
y = sin x
x
In this case,
Z π
2=
sin xdx
0
π sin 0 + sin π/4 sin π/4 + sin π/2 sin π/2 + sin 3π/4 sin 3π/4 + sin π
≈
+
+
+
4
2
2
2
2
√
π
= (2 + 2 2)
8
≈ 1.896
!
The trapezoid rule for uniform step size h = (b − a)/n can be written using
the notation
Z b
n−1
hX
f ( x) dx ≈
( f ( xk+1 ) + f ( xk ))
2
a
k =0
h
= ( f ( x0 ) + 2 f ( x1 ) + 2 f ( x2 ) + · · · + 2 f ( xn−1 ) + f ( xn ))
2
where x0 = a xn = b and xk = xk−1 + h for k = 1, . . . , n Use the trapezoid
rule to approximate the area under the curves below. What is the error?
(a) y = x2 from x = 0 to x = m using a step size of h = 1.
y = x2
(b) y = x2 from x = 0 to x = m using a step size h = m/n.
(c) y = 1/x from x = 1 to x = m using a step size h = 1.
x
Induction
1. Let F~ = hF0 , F1 , F2 , F3 , F4 , . . .i = h0, 1, 1, 2, 3, . . .i be the Fibonacci
sequence where Fn = Fn−1 + Fn−2 , F0 = 0 and F1 = 1. Prove Cassini’s
identity: For all n ≥ 1
Fn+1 Fn−1 − Fn2 = (−1)n
2. Let a > 1 be a positive integer. Pretend you want to divide n people
into some number of teams, each of size a or a + 1. Show that this is
possible provided n is larger than values in the Fibonacci polynomial
a2 − a − 1 = a(a − 1) − 1.
3. Find the coefficient of of xk in the expansion of ( x + 1/x)n .
4. Show that (see problem 1e)
n ( )
X
n
k =0
k
xk = xn
Relations
1. Identify the relations on the set of bits B = {0, 1} that are partial orders
and those that are equivalence relations.
2. Let F = { f : N → N} be the set of function from the natural numbers to
the natural numbers. Let f , g ∈ F and write f = O(g), saying “ f is big-O
of g,” if
(∃c ∈ N)(∃N ∈ N)(∀n ≥ N )( f (n) ≤ cg(n))
(a) Show that big-O is reflexive and transitive.
(b) Write f = Ω(g) and say “ f is big-Omega of g” if f = O(g) and
g = O( f ).
(c) Loosen the antisymmetry rule to conclude big-O is a loosened partial
order on the set F.
3. Consider the rules of “Rock-Paper-Scissors-Lizard-Spock”
• Scissors cuts paper
• Scissors decapitates lizard
• Paper covers rock
• Lizard eats paper
• Rock crushes lizard
• Paper disproves Spock
• Lizard poisons Spock
• Spock vaporizes rock
• Spock smashes scissors
• Rock crushes scissors
(a) Draw a graph which shows the rules of “Rock-Paper-Scissors-LizardSpock”
(b) Construct an adjacency matrix which shows the rules of “Rock-PaperScissors-Lizard-Spock”
24
discrete mathematics homework
(c) Is the game a partial order? Why or why not?
(d) Is the game an equivalence? Why or why not?
4. A candy machine takes nickels, dimes and quarters. When 50¢ is deposited a candy bar is dispensed along with change if necessary. The
diagram below shows the transitions from state-to-state as money is put in
the machine.
25
25
25
10
30
10
5
5
0
5
10
5
10
5
10
5
15
5
25
20
10
10|25
40
5
5
5
45
10
25
10
25
10
35
25
25
25
(a) Say that two paths p0 and p1 in the transition graph are path equivalent they start in the same state and end in the same state. Prove path
equivalence is an equivalence relation.
(b) Say state s precedes state t if there is a path from s to t. (Include
trivial paths from a state to itself on input of 0¢, so that state s precedes
itself). Prove precedes is a partial order.
5. Let a, b ∈ Z. What do the following mean? Discuss the equivalence
classes of integers that the relations define.
(a) a ≡ b (mod 2)
(b) a ≡ b (mod 1)
(c) a ≡ b (mod 0)
6. Let f : X → Y be a function. Let a, b ∈ X, and write a ≡ b if
f (a) = f (b). Show that ≡ is an equivalence relation.
5|10|25
50
Functions
1. What function does the following recursive function, written in a pseudocode, compute?
int f(int x, int [] a) {
// function f accepts input : integer x and
// integer array a[].
// f computes a value and returns it.
if (a. length () == 0) return 0;
else if (a. length () == 1) return a[0];
else return f(x, (a[0]*x+a[1] ,a.tail (). tail ()));
}
2. You have a picture that is 5” × 3”. You want to discretize it into an image
of 1920 × 1080 pixels. Into which pixel ( p x , p(y ) will ( x, y) fall, for
0 ≤ x ≤ 5, and 0 ≤ y ≤ 3?
You may want to start by studying the smaller 19 × 13 grid below.
(5, 3)
0
2
4
6
8
10
12
(0, 0)
0
2
4
6
8
10
12
14
16
18
3. Pretend you want to draw a straight line from pixel ( p x , py ) to pixel
(q x , qy ). Which pixels would you light?
Numbering & Naming Systems
1. How would you define a duovigesimal (base 32) number system? Discuss
the alphabet and modeling natural numbers, integers, and rational numbers
in fixed point and floating point notations. What is the compression in
string length when representing numbers in base 32 over base 2?
2. The powers of 2 form a basis for the natural numbers. Any natural number
n ≥ 0 can be written as the sum of powers of 2.


m−1
X


k

bk 2 
(∀n ∈ N)(∃bk ∈ B, k = 0, 1, . . . , m − 1) n =
k =0
Prove this.
3. The Fibonacci numbers are another interesting base for the natural numbers. That is,


m−1
X


(∀n ∈ N)(∃bk ∈ B, k = 0, 1, . . . , m − 1) n =
bk Fk 
k =0
Prove this.
4. I got this one from Click & Clack, the Tappet Brothers: Given $1000 in $1
dollar bills and 10 envelops, distribute the money among the envelops so
that you can give out any dollar amount from $1 to $1000.
5. Explain how you would define a three’s complement number system
to represent integers. What is the major irritation when the base for a
complement number system is not even?
6. Let r be a rational number so that r can be written as a fraction p/q in
lowest terms, that is, gcd( p, q) = 1.
(a) Show that if q = 2k for some integer k, then r can be expressed as a
finite binary string. Use fixed point notation.
(b) Show that if q , 2k for any integer k, then r cannot be expressed as a
finite binary string, but r can be written as a repeating binary pattern of
finite length.
28
discrete mathematics homework
7. Write 0.1, 0.2, and 0.3 in fixed point notation as repeating binary patterns. Use your ability to sum an infinite geometric series 5 to show your
answers are correct.
5
Infinite series is not taught in this class.
Series is a topic in calculus, but you can
guess, know, or look up geometric series.
8. In class we’ve used an 8-bit pidgin floating point notation to explain basic
concepts in modeling real numbers by a finite set of rational numbers.
Modern floating point units are expected to implement the IEEE 754
standard for binary floating-point arithmetic. A single precision floating
point numbers are represented in 32 bits: 1 sign bit, 8 exponent bits, and
23 fraction bits.
s
e7
e6
e5
e4
e3
e2
e1
byte 0
e0
f−1
f−2
f−3
f−4
f−5
f−6
f−7
byte 1
f−8
f−9
f−10
f−11
f−12
f−13
f−14
f−15 byte 2
f−16
f−17
f−18
f−19
f−20
f−21
f−22
f−23 byte 3
• The exponent e = (e7 · · · e0 )2 is biased with b = 127, giving a range
from e = 0 − 127 = −127 to e = 255 − 127 = 128.
– Normalized numbers x have exponent e in the range
emin = (0000 0001)b=127 = −126
to emax = (1111 1110)b=127 = 127
In this case,
x = (−1) s × (1. f )2 × 2e
– When e = −127 the numbers are denormalized or subnormal. When
the fraction bits are not all 0’s, the bit string represents plus or minus
0.
1 (0000 0000) (0000 0000 0000 0000 0000 000) = −0
0 (0000 0000) (0000 0000 0000 0000 0000 000) = +0
Otherwise, the bit string represents a small fraction.
1 (00000000) ( xxxx xxxx xxxx xxxx xxxx xxx) = −0. f × 2emin
0 (00000000) ( xxxx xxxx xxxx xxxx xxxx xxx) = +0. f × 2emin
– When e = 128 the bit string encodes ±∞
1 (1111 1111) (0000 0000 0000 0000 0000 000) = −∞
0 (1111 1111) (0000 0000 0000 0000 0000 000) = +∞
or Not a Number (NaN) numbers where the fraction is not all 0’s
1 (11111111) ( xxxx xxxx xxxx xxxx xxxx xxx) = NaN
0 (11111111) ( xxxx xxxx xxxx xxxx xxxx xxx) = NaN
numbering & naming systems
• Rounding occurs whenever an operation produces a result that cannot
be represented exactly in floating point notation. The default is to round
to the nearest floating point value.
(a) What is the single precision floating point approximation to 0.1, 0.2,
and 0.3?
(b) How are the single precision floating point approximations of 0.1 and
−0.2 stored as 4 byte hexadecimal strings of length 8?
(c) What are the smallest and largest positive normalized single precision
floating point numbers?
(d) What are the smallest and largest positive subnormal single precision
floating point numbers?
(e) Given that normalized single precision floating point numbers have
24 bits of precision, about how many decimal digits of accuracy can
you expect? Double precision uses 53 bits. How many decimal digits of
accuracy does this give?
29
Combinatorics & Number Theory
1. In how many ways can you find n positive integers that add up to m?
(Take the order of the terms into account, for example, 1 + 2 + 4 is different
than 2 + 4 + 1.)
2. An n-legged Alien has a drawer full of socks, each of which is red, white,
or blue, and there are at least n socks of each color. The Alien pulls out
one sock at a time without looking. How many socks must the Alien
remove from the drawer to be certain there will be n socks of the same
color?
Consider the problem again, but suppose there are m sock colors.
3. In the clock game Alice and Bob both start at 12 o’clock. During a move
Alice moves 5 hours clockwise on the clock-face and Bob moves 9 hours
counterclockwise. How many moves will it take before Bob and Alice
stop on the same hour?
What if the game is changes so that Alice moves a hours clockwise and
Bob moves b hours counterclockwise?
n−1
4. Let m = pe00 pe11 · · · pen−1
be the prime factorization of m. How many
positive divisors does m have?
5. Back in the day men always wore hats and when they went to a club they
would have a hat-check girl store their hats. If n men checked hats, in how
many ways can they be returned such that no man gets back his own hat?
6. The rumor is that Blaise Pascal was very interested in gambling and
developed many of his ideas about combinatorics from that.
(a) In how many five card hands can be dealt from a standard deck of 52
cards?
(b) In how many can each of the following hand be dealt?
i. A pair, for example
discrete mathematics homework
2
♥
J
♦
♦ ♦
♦
♦ ♦
♥
♥
♦
♥
♥
5
♠
5
♥
♦
♠
J
A
A
♦
♥
2
♠
A
A
ii. Two pairs, for example
2
K
♦
♥
♣
♥
♦
♥
♦
♥
♦
♥
2
2
♠
2
♥
♥
♠
♣
A
A
K
♠
A
A
iii. Three of a kind, for example
7
A
♥
♣
♥
♣
♦ ♦
♦
♦ ♦
♦ ♦
♥
♣
7
♠
Q
♦
♣
♦
♠
♣
A
A
Q
♠
A
A
A
iv. A straight, for example
5
♥
♥ ♥
♦ ♦
♦
♦ ♦
♥ ♥
7
♦
♣
♣ ♣
♣
♣ ♣
♣ ♣
♦ ♦
♦ ♦
♦ ♦
♦
6
♠
♠
♠
6
♦
♦
♠
♣
4
3
7
♥
5
♠
4
3
v. A flush, for example
5
10
♠
♠
♠ ♠
♠
♠ ♠
♠ ♠
♠
♠ ♠
♠
10
♠ ♠
♠
♠ ♠
♠
5
♠
3
vi. A full house, for example
K
♠
J
♠
♠
♠
J
♠
♠
♠
♠
3
K
32
combinatorics & number theory
♥
♦ ♦
♦
♦ ♦
♦ ♦
♣
♥ ♥
♥
♥ ♥
♥ ♥
♦
♥
7
♦
♣
7
♥
♣
A
7
♠
7
A
♥
♠
♥
A
A
♠
A
A
vii. Four of a kind, for example
A
A
♣
♣ ♣
♣ ♣
♣ ♣
♦
♦
♥
♣
♥
♣
A
A
♠
6
♦
♣
♣
♥
♠
6
A
A
♠
A
A
viii. Straight flush, for example
5
♠
♠ ♠
♠
♠ ♠
♠ ♠
♠
♠ ♠
♠ ♠
♠ ♠
♠ ♠
♠
♠ ♠
♠ ♠
♠
♠ ♠
7
♠
♠
6
♠
♠
♠
6
♠
♠
4
♠
7
3
♠
5
♠
4
3
ix. Royal flush, for example
10
Q
J
♥
A
K
♥
♥
♥
♥
♥ ♥
♥
♥ ♥
♥ ♥
♥
♥ ♥
♥
♥
Q
♥
♥
J
10
4
8
7
And a full binary tree
♥
487
K
A (not full) binary tree
♥
7. Binary trees are basic data structures. A full binary tree is one where each
internal node has exactly two children.
A
(c) What is the probability of each of the above hands?
33
34
discrete mathematics homework
1234
34
12
1
2
3
4
(a) Let T n be a full binary tree with n internal nodes. How many edges
does T n have?
(b) Let T n be a full binary tree with n internal nodes. How many leafs
does T n have?
(c) Let Cn be the number of full binary trees with n internal nodes. What
is a recurrence equation for Cn ?
(d) Let Cn be the number of full binary trees with n internal nodes. What
is a formula, involving binomail coefficients, for Cn ?
Cn is called a Catalan number.
Proofs
1. Let a and b be positive integers. Prove that g = gcd(a, b) is the smallest
positive value of as + bt as s and t range over the integers.
2. A real number r is called sensible if there exist positive integers a and b
√
√
such that a/b = r. For example, setting a = 2 and b = 1 shows that 2
√
3
is sensible. Prove that 2 is not sensible. 6
3. Pretend you have n pigeons to put in m pigeonholes.
(a) Prove there must be a pigeonhole with n/m or more pigeons.
(b) Prove there must be a pigeonhole with n/m or fewer pigeons.
4. Place a knight on each square of a 7 × 7 chessboard. ls it possible for each
knight to simultaneously make a legal move?
6
This problem came from MIT course
6.042J/18.062J, by Prof. Albert R. Meyer
and Prof. Ronitt Rubinfeld
proofs
37
Index
Axiom of Choice, 11
Big-Oh Notation, 23
Binary addition, 7
Binary tree, 33
Full, 33
Binomial Theorem, 21
Bits
Carry In, 7
Carry Out, 7
Boole, George, 7
Boolean Functions, 7
n-Input, m-Output, 7
Quasi-Boolean, 8
Fibonacci Basis, 27
Fibonacci Polynomial, 21
Full adder, 7
Functions, 25
Pascal,Blaise, 31
Permutations, 16
Pigeons and Pigeonholes, 35
Proofs, 35
Graphs of Quantified Predicate, 9
Greatest Common Divisor, 35
Growth of Arithmetic and Geometric
Sequences, 13
Rational Numbers, 27
Base Two, 27
Fixed Point Representation, 28
Floating Point Number Representation,
28
Recursion, 15
Relations, 23
Relationship between powers of x, Falling
Factorials, and Second Stirling
Numbers, 21
Homework
Calculating Grades, 6
IEEE 754, 28
Induction, 21
Cantor set, 12
Cantor, Georg, 12
Cassini’s Identity, 21
Catalan numbers, 34
Click & Clack, the Tappet Brothers, 27
Combinatorics, 31
Complement Number Systems, 27
Logic, 7
Quantifying Predicates, 9
Equivalence Relations, 23
Eulerian Numbers, 16
Partial Orders, 23
Partitions of a Positive Integer, 15
Making teams, 21
Number Systems, 27
Number Theory, 31
Sensible Numbers, 35
Sequences, 13
Ascents, 16
Sets, 11
Bits B = {0, 1}, 23
Integers Z = {0, ±1, ±2, . . .}, 24
Natural Numbers N = {0, 1, 2, . . .}, 23
Summations, 19
Trapezoid Rule, 19