Section 5.1
115
Mathematical Induction
CHAPTER 5
Induction and Recursion
SECTION 5.1
Mathematical Induction
Important note about notation for proofs by mathematical induction: In performing the inductive
step, it really does not matter what letter we use.
Week 1 Homework :13
:Section 11
Let p and q be the propositions The election is decided and The votes have .10
been counted, respectively. Express each of these compound propositions as an
.English sentence
.: The election is not decided, and the votes
Assignment week 12
Section A
Determine whether each of these statements is true or false
1) Suppose that the relation R on a set is represented by the matrix
[ ]
1 1 0
M R= 1 1 1
0 1 1
Then R is reflexive, symmetric but not antisymmetric. T
2) The transi
152
Chapter 6
Counting
CHAPTER 6
Counting
SECTION 6.1
The Basics of Counting
2. By the product rule there are 27 37 = 999 oces.
4. By the product rule there are 12 2 3 = 72 dierent types of shirt.
6. By the product rule there are 4 6 = 24 routes.
8. There
Section 1.1
Propositional Logic
1
CHAPTER 1 The Foundations: Logic and Proofs
SECTION 1.1 Propositional Logic
2. Propositions must have clearly defined truth values, so a proposition must be a declarative sentence with no free variables. a) This is not a
258
Chapter 10
Graphs
CHAPTER 10
Graphs
SECTION 10.1
Graphs and Graph Models
2. a) A simple graph would be the model here, since there are no parallel edges or loops, and the edges are
undirected.
b) A multigraph would, in theory, be needed here, since th
Section 9.1
231
Relations and Their Properties
CHAPTER 9
Relations
SECTION 9.1
Relations and Their Properties
2. a) (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 2), (2, 4), (2, 6), (3, 3), (3, 6), (4, 4), (5, 5), (6, 6)
b) We draw a line from a to
38
Chapter 2
Basic Structures: Sets, Functions, Sequences, Sums, and Matrices
CHAPTER 2 Basic Structures: Sets, Functions, Sequences, Sums, and Matrices
SECTION 2.1 Sets
2. There are of course an infinite number of correct answers. a) cfw_ 3n  n = 0, 1,
178
Chapter 7
Discrete Probability
CHAPTER 7
Discrete Probability
SECTION 7.1
An Introduction to Discrete Probability
2. The probability is 1/6 0.17, since there are six equally likely outcomes.
4. Since April has 30 days, the answer is 30/366 = 5/61 0.08
Section 4.1
Divisibility and Modular Arithmetic
87
CHAPTER 4
Number Theory and Cryptography
SECTION 4.1
Divisibility and Modular Arithmetic
2. a) 1  a since a = 1 a.
b) a  0 since 0 = a 0.
4. Suppose a  b , so that b = at for some t , and b  c, so tha
66
Chapter 3
Algorithms
CHAPTER 3
Algorithms
SECTION 3.1
Algorithms
2. a) This procedure is not nite, since execution of the while loop continues forever.
b) This procedure is not eective, because the step m := 1/n cannot be performed when n = 0, which wi
:
.
:
)
(
:
.
.
:
:
.
(
)
.
Universal Knowledge Solutions s.a.l.
1
:
:
F
T
:
"2+2=4" : 1
"2+2=5" : 2
(x=1,y=1) y x
" x+y >0" : 3
(x=1,y=1)
:
x
(r
) x2 > 4
(Q
) x < 2
(p
:
If p or Q then r
.
:
(Q
)
(p
)
.(r
)
:
If p or Q then r
:
If not r then not p and
Lets get started with.
Logic!
Fall 2002
CMSC 203  Discrete
1
Logic
Crucial for mathematical reasoning
Used for designing electronic circuitry
Logic is a system based on propositions.
A proposition is a statement that is either
true or false (not both).
196
Chapter 8
Advanced Counting Techniques
CHAPTER 8
Advanced Counting Techniques
SECTION 8.1
Applications of Recurrence Relations
2. a) A permutation of a set with n elements consists of a choice of a rst element (which can be done in n
ways), followed b
Assignment of week 14/ Math 150
Section 1: (True or False Questions)
Section 1
1
A connected graph without any simple circuit is called a tree. T
2 A tree with n vertices has n1 edges. T
3 If desecendents and ancestors of a node is empty then this node h
Assignment week 13
1. Determine whether each of these statements is true or
false.
(a)
An undirected graph has an even number of
vertices of odd degree. T
(b)
A complete graph on n vertices, denoted by Kn , is
a simple graph that contains exactly one edge
Assignments for week11
Question Number(1):
Determine whether each of these statements is true or false?
pn=(1.11) p
1)The recurrence relation
is a linear homogeneous
n1
T
recurrence relation of degree one.
2)The recurrence relation
an= a
n1
an =6 an19 an2
Assignment week 8
Section A
Determine whether each of these statements is true or false
1) The first five Fibonacci numbers are: 1,2,3,5,8. T
2) A simple polygon with n sides, where n is an integer with
triangulated into n2 triangles.
f ( 0 )=0, f ( n )=2
Saudi Electronic University
College of Computing and Informatics
Assignment of week 6/ Math 150
Section A: (True or False Questions)
Section 1
1.
2.
3.
4.
f(x)=x+x3 is O(x3)
3644 mod 645 equal 35
If f1(x) and f2(x) are both O(g(x). Then (f1+f2)(x) is O(g(
Muhannad alloush 130081052
Q.1/
A.
B.
C.
D.
E.
False
True
False
True
False
Q.2/
a. Universal quantifier is denoted by
B
b. Existential quantifier is denoted by
A
c. The statement : _ xP(x) is equivalent to
D
d. The statement : x(P(x)  Q(x) is equivalent
Assignment of week 3/ Math 150
Muhannad alloush 130081052
Section A: (True or False Questions)
Section 1
1The OR function is Boolean multiplication and the AND function is Boolean addition. F
2In Boolean algebra theorem is very useful for minimizing log
THE ANSWERS
:Q1
12345
False
FALSE ( because the statement is false when is true and is false )
TRUE
FALSE ( because ~ is a tautology )
FALSE ( because the biconditional statement is false when one of or is false)

:Q2
12345
c
A
A
A
D

Q3:
1 Show t
Assignment week 5
Section A
Determine whether each of these statements is true or false
1)
cfw_ x cfw_x , cfw_x . T
2) The function
f ( x )=
x+ 1
x+ 2 is not a bijection from R cfw_ 2 to R cfw_1. F
3) Suppose that A is an n n matrix where n is a positi
Assignment of week 7/ Math 150
Section 1: (True or False Questions)
Section 1
F
1/ 7/3 is a integer.
2
Let a,b,c be integers where a 0 then . if a/b and a/c , then a/(a+b) .
3
In the binary notation each digit is either a , 0 or a, 1.
T
4
Convert ( 101011
REPBLICA DE ANGOLA
REGIO ACADMICA III
UNIVERSIDADE 11 DE NOVEMBRO
ESCOLA SUPERIOR POLITCNICA DO ZAIRE (SOYO)
Cadeira: MECANICA DOS FLUIDOS 1.
Tempo de aula: 20 horas
Docente: M.Sc. Noidys Quirs Martn
Curso: OMI IAno/2015
Conteudos a desenvolver:
Enunciar
Week 10
Q 1:Determine whether each of these statements is true
or false:
1The number of function from a set with 3 elements to a set with 6
F
elements is 18.
/ Sol : A function corresponds to a choice of one of the n elements in the
codomain for each of
Week
1
2
Title
Orientation
Logic and Proofs I
3
Boolean Algebra
4
Logic and Proofs II
5
Sets, Functions,
Sequences, Sums, and
Matrices
Algorithms
6
7
8
9
10
11
Number Theory and
Cryptography
Midterm Exam
Induction and
Recursion
Counting
12
Advanced Counti
CSTSSEUKSA
Discrete mathematics Math150
Assignment 1
1st semester 20162017
Section I
(1 mark for each)
Determine whether the statement is true (T) or false (F).
1) The statement where are you? is proposition.
2) The conjunction of propositions p and
Show All Solutions
Rosen, Discrete Mathematics and Its Applications, 7th edition
Extra Examples
Section 2.4Sequences and Summations
Page references correspond to locations of Extra Examples icons in the textbook.
p.160, icon at Example 11
#1. Find a rule
Show All Solutions
Rosen, Discrete Mathematics and Its Applications, 7th edition
Extra Examples
Section 1.4Predicates and Quantiers
Page references correspond to locations of Extra Examples icons in the textbook.
p.38, icon at Example 3
#1. Let P (x) be