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### Klp Mishra

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# Klp Mishra

By | April 2013
Page 1 of 755
THEORY OF COMPUTER SCIENCE
Automata, Languages and Computation
THIRD EDITION

K.l.P. MISHRA
Formerly Professor
Department of Electrical and Electronics Engineering
and Principal/ Regional Engineering College
Tiruchirapal/i

N. CHANDRASEKARAN
Professor
Department of Mathematics
St. Joseph/s College
Tiruchirapalli

Prentice' Hall of India [P[?lmGJD@ LsOWJov8d]
New Delhi - 110 '001
2008

Contents

Preface

ix

Notations

Xl

1.

PROPOSITIONS AND PREDICATES
1.1 Propositions (or Statements)
1
1.1.1 Connectives (Propositional Connectives
or Logical Connectives)
2
1.1.2 Well-formed Formulas
6
1.1.3 Truth Table for a Well-formed Formula
7
1.1.4 Equivalence of Well-formed Formulas
9
1.1.5 Logical Identities
9
1.2 Normal Forms of Well-formed Formulas
11
1.2.1 Construction to Obtain a Disjunctive Normal
Form of a Given Formula
II
1.2.2 Construction to Obtain the Principal
Disjunctive Normal Form of a Given Formula
1.3 Rules of Inference for Propositional Calculus
(Statement Calculus)
15
1.4 Predicate Calculus
19
1.4.1 Predicates
19
1.4.2 Well-formed Formulas of Predicate Calculus
1.5 Rules of Inference for Predicate Calculus
23
1.6 Supplementary Examples
26
Se(f-Test
Exercises

31
32

iii

1-35

12

21

iv

J;J

Contents

2. MATHEMATICAL PRELIMINARIES
2.1 Sets, Relations and Functions
36
2.1.1 Sets and Subsets
36
37
2.1.2 Sets with One Binary Operation
2.1.3 Sets with Two Binary Operations
39
2.1.4 Relations
40
43
2.1.5 Closure of Relations
2.1.6 Functions
45
2.2 Graphs and Trees
47
2.2.1 Graphs
47
2.2.2 Trees
49
2.3 Strings and Their Properties
54
2.3.1 Operations on Strings
54
2.3.2 Terminal and Nonterrninal Symbols
56
2.4 Principle of Induction
57
2.4.1 Method of Proof by Induction
57
58
2.4.2 Modified Method of Induction
2.4.3 Simultaneous Induction
60
61
2.6 Supplementary Examples
62
Self-Test
Exercises

66
67...

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