Table Of Content2019 First Edition
DDDiiissscccrrreeettteee
MMMaaattthhheeemmmaaatttiiicccsss
engineering handbook Arjun Singh
GRAB YOUR COPY NOW !!!
CONTENTS
1
THE FOUNDATIONS : LOGIC AND
CH APTER . .
METHOD OF PROOFS
1.1 PROPOSITIONAL LOGIC 1
SOLVED EXERCISE - 1.1 8
1.2 PROPOSITIONAL EQUIVALENCE 27
SOLVED EXERCISE - 1.2 28
1.3 PREDICATES AND QUANTIFIER 38
SOLVED EXERCISE - 1.3 43
1.4 NESTED QUANTIFIERS 58
SOLVED EXERCISE - 1.4 61
1.5 RULES OF INFERENCE 76
SOLVED EXERCISE - 1.5 78
1.6 INTRODUCTION TO PROOF 88
SOLVED EXERCISE-1.6 91
1.7 PROOF METHOD AND STRATEGY 96
SOLVED EXERCISE-1.7 98
2
SEQUENCE, INDUCTION AND RECURSION
CH APTER . .
2.1 SEQUENCES AND SUMMATIONS 103
SOLVED EXERCISE -2.1 (2.4) 108
2.2 MATHEMATICAL INDUCTION 117
SOLVED EXERCISE - 2.2. (4.1) 120
2.3 STRONG INDUCTION 136
SOLVED EXERCISE -2.3 (4.2) 140
2.4 RECURSIVELY DEFINED FUNCTION 146
SOLVED EXERCISE- 2.4(4.3) 156
2.5 RECURSIVE ALGORITHMS 164
SOLVED EXERCISE- 2.5 (4.4) 167
2.6 PROGRAM CORRECTNESS 169
SOLVED EXERCISE-2.6 (4.5) 173
3
ADVANCE COUNTING TECHNIQUES
CH APTER . .
3.1 RECURRENCE RELATION 177
SOLVED EXERCISE-3.1(6.1) 181
3.2 SOLVING RECURRENCE RELATION 190
SOLVED EXERCISE - 3.2 (6.2) 197
3.3 GENERATING FUNCTION 201
SOLVED EXERCISE-3.3(6.4) 209
3.4 INCLUSION AND EXCLUSION PRINCIPLE 221
SOLVED EXERCISE - 3.4 (6.5) 223
3.5 APPLICATION OF INCLUSION-EXCLUSION PRINCIPLE 227
SOLVED EXERCISE -3.5 (6.6) 229
4
RELATIONS
CH APTER . .
4.1 RELATION AND THEIR PROPERTIES 232
SOLVED EXERCISE-4.1 (7.1) 241
4.2 n-ary RELATION 249
SOLVED EXERCISE-4.2(7.2) 251
4.3 REPRESENTATION OF RELATIONS 253
SOLVED EXERCISE-4.3 (7.3) 258
4.4 CLOSURE OF RELATIONS 265
SOLVED EXERCISE-4.4 (7.4) 276
4.5 EQUIVALENCE RELATIONS 283
SOLVED EXERCISE - 4.5 (7.5) 290
4.6 PARTIAL ORDERING RELATIONS 293
SOLVED EXERCISE - 4.6 (7.6) 294
5
GRAPH THEORY
CH APTER . .
5.1 INTRODUCTION TO GRAPHS 301
SOLVED EXERCISE - 5.1 (8.1) 304
5.2 GRAPH TERMINOLOGY 308
SOLVED EXERCISE - 5.2 (8.2) 319
5.3 MATRIX REPRESENTATION OF GRAPHS 323
SOLVED EXERCISE - 5.3 (8.3) 328
5.4 PATHS AND CONNECTIVITY 334
SOLVED EXERCISE - 5.4 (8.4) 341
5.5 EULERIAN AND HAMILTONIAN GRAPH 345
SOLVED EXERCISE - 5.5 (8.5) 353
5.6 SHORTEST PATH PROBLEMS 357
SOLVED EXERCISE - 5.6 (8.6) 364
5.7 PLANAR GRAPHS 368
SOLVED EXERCISE - 5.7 (8.7) 373
5.8 GRAPH COLORING 377
SOLVED EXERCISE - 5.8 (8.8) 386
6
TREES
CH APTER . .
6.1 INTRODUCTION TO TREES 390
SOLVED EXERCISE - 6.1 (9.1) 397
6.2 APPLICATION OF TREES 400
SOLVED EXERCISE-6.2(9.2) 404
6.3 TREE TRAVERSAL 413
SOLVED EXERCISE - 6.3 (9.3) 415
6.4 SPANNING TREES 419
SOLVED EXERCISE - 6.4 (9.4) 421
6.5 MINIMAL SPANNING TREE 425
7
ALGEBRAIC STRUCTURE
CH APTER . .
7.1 INTRODUCTION TO GROUP 429
7.2 SUB GROUPS 434
7.3 COSETS AND LAGRANGE’S THEOREM 435
7.4 PERMUTATION GROUP 440
7.5 GROUP CODES 443
7.6 HOMOMORPHISM AND ISOMORPHISM OF GROUPS 445
7.7 NORMAL SUB GROUPS 446
7.8 RINGS, INTEGRAL DOMAINS, AND FIELDS 447
7.9 RING HOMOMORPHISM 450
SOLVED EXERCISE - 7(10) 453
8
LATTICES & BOOLEAN ALGEBRA
CH APTER . .
8.1 LATTICE & ALGEBRAIC SYSTEMS 469
8.2 PRINCIPLE OF DUALITY 472
8.3 BASIC PROPERTIES OF ALGEBRAIC SYSTEMS DEFINED BY LATTICES 472
8.4 BOOLEAN LATTICES AND BOOLEAN ALGEBRAS 474
8.5 BOOLEAN FUNCTION AND BOOLEAN EXPRESSIONS 475
SOLVED EXERCISE - 8 (11) 477
ppp
THE FOUNDATIONS : LOGIC
AND METHOD OF PROOFS
1.1 PROPOSITIONAL LOGIC
In this chapter we shall study mathematical logic, which is the basics of all kinds of
reasoning. Mathematical logic has two aspects. On one hand it is analytical theory of art
of reasoning whose goal is to systematize and codify principles of valid reasoning. It may
be used to judge the correctness of statements which make up the chain. In this aspect
logic may be called 'classical' mathematical logic. The other aspect of mathematical logic
is inter-related with problems relating the foundation of Mathematics. G. Frege (1884-
1925) developed the idea, regarding a mathematical theory as applied system of logic.
Principles of logic are valuable to problem analysis, programming and logic design.
Proposition
A number of words making a complete grammatical structure having a sense and
meaning in logic or mathematics is called as entence. This assertion may be of two types
- declarative and non-declarative. A Proposition or Statement is a declarative sentence
that is either true of false, For example, “Three plus four equals seven” and “Three plus
four equals eight” are both statements, the first because it is true and the second because
it is false. Similarly “x + y > 2” is not a statement because for some values of x and y the
sentence is true, whereas for others it is false. For instance, if x = 1 and y = 2, the sentence
is true, if x = –3 and y = 1, this is false. The truth or falsity of a statement is called tirtsu th
value. Since only two possible truth values are admitted this logic is sometimes called
two - valued logic. Questions, exclamations and commands are not propositions. For
examples, consider the following sentences.
(a) The sun rises in the north.
(b) 12 + 4 = 16
(c) {5,6}(cid:204){7,6,5}
(d) Do you speak Hindi?
(e) 4 – x = 8
(f) Close the door,
(g) What a hot day!
(h) We shall have chicken for dinner.
The sentences (a), (b) and (c) are statements, the first is false and second and third are
2 Discrete Mathematical Structure
true.
(d) is a question, not a declarative sentence, hence it is not a statement.
(e) is a declarative sentence, but not a statement, since it is true or false depends on
the value of x.
(f) is not a statement, it is a command.
(g) is not a statement, it is exclamation.
(h) is a statement since it is either true or false but not both, although one has to wait
until dinner to find out if it is true or false.
It is necessary to represent simple statements by letter p, q, r........... known as
proposition variables (Note that usually a real variable is represented by the symbol x.
This means that x is not a real number but can take a real value. Similarly, a propositional
variable is not a proposition but can be replaced by a proposition) Propositional variables
can only assume two values, true or false. There are also twop ropositional constants, T
and F, that represent true and false, respectively. If p denotes the proposition “The sun sets
in the west”, then instead of saying the proposition “ The sun sets in the west:” is false,
one can simply say the value of p is F.
Compound Proposition
A proposition consisting of only a single propositional variable or a single
propositional constant is called anp rimary, (primitive)p roposition or simply proposition;
that is they can not be further subdivided. A proposition obtained from the combinations
of two or more propositions by means of logical operators or connectives of two or more
propositions or by negating a single proposition is referred tom olecular or composite or
compound proposition.
Connectives
The words and phrases (or symbols) used to form compound propositions are called
connectives. There are five basic connectives called Negation, Conjunction, Disjunction,
Implication or Conditional and Equivalence or Biconditional. The following symbols are
used to represent connectives.
Symbol Connective Nature of the compound Symbolic Negation
used word statement formed by the from
connective
~, (cid:216) not Negation ~p ~(~p) = p
(cid:217) and Conjuction p (cid:217) q (-p) (cid:218) (~q)
(cid:218) or Disjunction p (cid:218) q (-p) (cid:217) (~q)
(cid:222), fi if... ...... .then Implication (or Conditional) p (cid:222) q p (cid:217) (~q)
(cid:219),<--> if and only if Equivalence (or Bi- p (cid:219) q [p (cid:217) (~q)] (cid:218)
conditional) [ q (cid:217) (~p)]
Discrete Mathematical Structure 3
Negation
If p is any proposition, the negation of p, denoted by ~p or(cid:216) p and read as not p, is
a proposition which is false when p is true and true when p is false. Consider the statement
p: Paris is in England.
Then the negation of p is the statement ,
~p: It is not the case that Paris is in England.
Usually it is written as
~p: Paris is not in England.
Strictly speaking, not is not connective, since it does not join two statements and ~p
is not really a compound statement. However, not is a unary operation for the collection of
statements, and ~p is a statement if p is considered a statement.
Note: 1. The following propositions all have the same meaning:
p: All people are intelligent.
q: Every person is intelligent.
r: Each person is intelligent.
s: Any person is intelligent.
2. The negation of the proposition is
p: All students are intelligent.
~p: Some students are not intelligent.
~p: There exists a student who is not intelligent.
~p: There exist students who are not intelligent.
~p: At least are student is not intelligent.
The negation of
q: No student is intelligent, is
~q: Some students are intelligent.
Note that “No student is intelligent” is not the negation of p; “All students are intelligent”
is not the negation of q.
Example -1
Find the negation of the following statements.
Solution :
Statement Negation
a) p : You are my brother, ~p : You are not my brother.
b) q : Think twice before doing this, ~q : Don’t think twice before doing this.
c) r : There are many hurdles in this way. ~r : There is no hurdles in this way.
d) s : He slaps me very hard, ~s : He doesn’t slaps me hard.
