10.13
(CarbonFootprint Interface: Polymorphism)
Using interfaces, as you learned in this chapter, you can specify similar behaviors for
possibly disparate classes. Governments and companies worldwide are becoming
increasingly concerned with carbon footpri

Read books online java how to program 9th edition exercise solutions or download java how to program 9th edition exercise solutions for
free
JAVA HOW TO PROGRAM 9TH EDITION EXERCISE
SOLUTIONS PDF
Download : Java How To Program 9th Edition Exercise Solutio

p
~p
~(~p)
T
F
T
F
Discrete Mathematics
T
F
LECTURE #2
Truth Tables for:
1.
~pq
2.
~ p (q ~ r)
3.
(pq) ~ (pq)
Truth table for the statement form ~ p q
p
q
~p
T
T
F
~p
q
F
T
F
F
F
F
T
T
T
F
F
T
F
Truth table for ~ p (q ~ r)
Truth table for (pq) ~ (pq)
p
q

Discrete Mathematics
LECTURE #3
APPLYING LAWS OF LOGIC
Given any statement variables p, q, and r , a tautology t and a contradiction c, the following
logical equivalences hold.
1. Commutative laws:
p q q p and
p q q p
2. Associative laws:
(p q) r p (q r)

CSC102-Discrete Mathematics
LECTURE # 13
EXERCISE:
Suppose R and S are binary relations on a set A.
a. If R and S are reflexive, is R S reflexive?
b. If R and S are symmetric, is R S symmetric?
c. If R and S are transitive, is R S transitive?
SOLUTION:
a.

Discrete Mathematics
LECTURE # 9
SET IDENTITIES:
Let A, B, C be subsets of a universal set U.
1.
Idempotent Laws
a.
AA=A
b.
AA=A
2.
Commutative Laws
a.
AB=BA
b.
AB=BA
3.
Associative Laws
a.
A (B C) = (A B) C
b.
A (B C) = (A B) C
4.
Distributive Laws
a.
A

Discrete Mathematics
LECTURE # 1
Course Objective:
1.Express statements with the precision of formal logic
2.Analyze arguments to test their validity
3.Apply the basic properties and operations related to sets
4.Apply to sets the basic properties and oper

p
q
pq
p
q
T
T
T
T
T
T
F
F
T
F
F
T
T
F
T
F
F
T
F
F
Discrete Mathematics
LECTURE # 5
EXAMPLE
An interesting teacher keeps me awake. I stay awake in Discrete Mathematics class.
Therefore, my Discrete Mathematics teacher is interesting.
Is the above argument

CSC102 Discrete Mathematics
LECTURE # 11
ORDERED PAIR:
An ordered pair (a, b) consists of two elements a and b in which a is the
element and b is the second element.
The ordered pairs (a, b) and (c, d) are equal if, and only if, a= c and b = d.
Note that

CSC102-Discrete Mathematics
LECTURE # 12
REFLEXIVE RELATION:
Let R be a relation on a set A. R is reflexive if, and only if, for all a A,
(a, a) R. Or equivalently aRa.
That is, each element of A is related to itself.
REMARK
R is not reflexive iff there i

CSC102-Discrete Mathematics
LECTURE # 15
RELATIONS AND FUNCTIONS:
A function F from a set X to a set Y is a relation from X to Y that satisfies the following two
properties
1.For every element x in X, there is an element y in Y such that (x,y) F.
In other

Discrete Mathematics
LECTURE
A well defined collection of cfw_distinctobjects is called a set.
The objects are called the elements or members of the set.
Sets are denoted by capital letters A, B, C , X, Y, Z.
The elements of a set are represented by lower

Discrete Mathematics
LECTURE # 8
UNION:
Let A and B be subsets of a universal set U. The union of sets A and B is the set of
all elements in U that belong to A or to B or to both, and is denoted A B.
Symbolically:
A B = cfw_x U | x A or x B
EMAMPLE:
Let U

CSC102-Discrete Mathematics
LECTURE # 14
INVERSE OF A RELATION:
Let R be a relation from A to B. The inverse relation R-1 from B to A is defined
as:
R-1 = cfw_(b,a) BA | (a,b) R
More simply, the inverse relation R-1 of R is obtained by interchanging the e

Discrete Mathematics
LECTURE # 4
BICONDITIONAL
If p and q are statement variables, the biconditional of p and q is
p if, and only if, q and is denoted pq. if and only if abbreviated iff.
The double headed arrow " " is the biconditional operator.
TRUTH TAB

CSD101 - Discrete Structures
(Discrete Mathematics)
Fall 2016
Lecture - 9
Set Operations
Set Operations
Two sets can be combined in many different ways.
Set operations can be used to combine sets.
Union
Let A and B be sets.
The union of A and B, denot

Question 1.How Google rank pages?
The websites that Google ranks on the 1st page of its search results for any given search
term are the ones that they consider to be the most relevant and useful. They determine
which websites are the most useful and rele

1. aegis The ae in this word is pronounced /ee/. Say EE-JIS/, not /ay-jis/.
/ids/
aegis \-js also -\
2. archipelago The ch is pronounced with a /k/ sound. Say /AR-KI-PEL-A-GO/, not /arch-i-pela-go/.
kpel/
archipelago \r-k-pe-l-g, r-ch-\
3. arctic Note the

1. aegis The ae in this word is pronounced /ee/. Say EE-JIS/, not /ay-jis/. In mythology the aegis is associated
especially with the goddess Athene. It is her shield with the Gorgons head on it.
2. anyway The problem with this word is not so much pronunci

1
Almond: This dry fruit is pronounced as Ah-mund,
(pronunciation of u as in sun). L is silent. You should not
pronounce it as Aal-mund.
Oxford dictionary
NAmE /mnd/
Merriam Webster
nounalmond\ m nd, a, l, al\
1. Bury: It is pronounced as Be-ri, the same

Introduction to democracy
Introduction:Democracy has now become the universal term describing the best political system. We have
analyzed democracy in a different way with its history and effects. We tried to find the answer to
the question that how democ

C
CLAPTRIX
OCTET
CAUSES OF FAILURE OF
DEMOCRACY
Corruption.!
Hunger for power.!
Ego.!
These are such characteristics of a politician which are playing an important role for
corrupting and manipulating the entire democratic system, causing its failure.
Now

6.5.1 Moments of Smallest Eigenvalue (lm00) : AK)
Theorem 5.1 The pth moment of the smallest eigenvalue EULA00) may be
derived as [27]
K . . K_1 L1~K,1 K71 ' '
Wq<oo>=colz<1>'+Izsgn(a>nra,k>( 2 b)
ijzl
7 Kl i.
a: k: 5K 1)!Kzll +1w 1
(6.10)
ll~K71 l
where

The amount of income spent on housing is an important component of the cost of liv
Five
Years
Ago
Homeowner
Now
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
17
20
29
43
36
43
45
19
49
49
35
16
23
33
44
44
28
29
39
22
10
39
37
27
12
41
24
26
28
26
32

CSD101 - Discrete Structures
(Discrete Mathematics)
Fall 2016
Lecture - 13
Recursion
Recursive Objects
Recursion is the process of repeating items in a self-
similar way.
Sometimes it is difficult to define an object explicitly, but it
is easier to defi

CSD101 - Discrete Structures
(Discrete Mathematics)
Fall 2016
Lecture - 5
Rules of Inference
Rules of Inference
Rules of inference are templates for constructing valid
arguments.
Rules of inference are basic tools for establishing the truth
of statement

CSD101- Discrete Structures
(Discrete Mathematics)
Fall 2016
Lecture - 7
Introduction to
Proofs
Proofs
Proof:
A proof is a valid argument that establishes the truth of a
mathematical statement.
Proofs are essential in mathematics and computer
science.