Mathematical Foundation of Computer Science

Size: px

Start display at page:

Download "Mathematical Foundation of Computer Science"

Regina Moore
5 years ago
Views:

2 Mathematical Foundation of Computer Science

3 Mathematical Foundation of Computer Science Covers the syllabus of: IP University, Delhi; SRM University, Chennai; MTU Noida; GBTU Lucknow; VIT, Tamil Nadu; Madhya Pradesh Bhoj Open University; Punjabi University, Patiala; University of Jammu (For BCA); Bharathidasan University, Tiruchirappalli (For MSc (CS)); Institute of Advance Studies Education (Deemed University) For PGDCA); MDU, Rohtak (for BCA); Uttrakhand Technical University; Ghauhati University (for M.Sc (CS)); Sri Chandrasekharendra Saraswathi Viswa Mahavidyalaya, Enathur, Kanchipuram (For ME). Manisha Singhwal MA, Mathematics (Gold Medalist) MCA, M.Phil. Assistant Professor, Department of Computer Science Institute of Professional Excellence and Management Ghaziabad (U.P.) Asian Books Private Limited

4 Mathematical Foundation of Computer Science Manisha Singhwal Asian Books Private Limited Registered and Editorial Office 7/28, Mahavir Lane, Vardan House, Ansari Road, Darya Ganj, New Delhi Ph.: , , Fax: Website: Publishers First Edition 2013 ISBN All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording and/or otherwise, without the prior written permission of the publisher. Published by Kamal Jagasia for Asian Books Pvt. Ltd., 7/28, Mahavir Lane, Vardan House, Darya Ganj, New Delhi Typeset at: Sara Assignments, Shahdara, Delhi Printed at: Anubha Printers, Greater Noida Sales Offices New Delhi 7/28, Mahavir Lane, Vardan House, Ansari Road, Darya Ganj, New Delhi Ph.: , , Fax: , sales@asianbooksindia.com Bangalore 103, Swiss Complex, 33, Race Course Road, Bangalore Ph.: , , Fax: , asianbangalore@asianbooksindia.com Guwahati House No. 48, Lamb Road, Latashil, P.O. Latashil, Distt. Kamrup (Metro), Guwahati , Assam Ph.: , , asianguwahati@asianbooksindia.com Kolkata 10 A, Hospital Street, Kolkata Ph.: , , , Fax: , customer.servicekolkatta@asianbooksindia.com Mumbai 103, Blackie House, Walchand Hirachand Marg, Opp. GPO, VT, Mumbai Ph.: , , , asianbooksvt@asianbooksindia.com Resident Offices Chennai Shop No. 19, First Floor, Real Regency Complex, 102, Pycrofts Road, Royapettah, Chennai Ph.: , asianchennai123@gmail.com Hyderabad , Street No. 7, Vittalwadi, Narayanguda, Hyderabad Ph.: , asianhyderabad@asianbooksindia.com Pune Shop No. 5-8, G.F. Shaan Brahma Complex, Near Ratan Theater, Budhwarpeth, Pune , Ph: , asianpune@asianbooksindia.com

5 Dedicated to My Parents

6 Preface Mathematical Foundations of Computer Science explains the Fundamental Concepts in Mathematics. It can be used by the students in Computer Science as an introduction to the underlying ideas of mathematics for computer science. This book is designed to provide an introduction to some Fundamental Concepts in Discrete Mathematics and Automata. Mathematical Foundation of Computer Science is a compulsory paper in most computing program universities of our country. It covers Set Theory, Relation, Function Mathematical Induction, Recurrence Relation generating Function, Algebraic structure, Lattices, Graph Theory, Finite Automation, Regular Expression, Turing Machine, and Formal Language. Each topic in the book has neem treated as easy manner. A set of exercise (with answers) has also been given at then end of each chapter to test the students s comprehension. The proofs of various Theorems and Examples have been given with enough details. Numerous Solved Examples have been included in the book to help the students in understanding various concepts easily. I have tried my best to keep the book free from misprints. I will be grateful to the readers who point out errors and omissions which in spite of all case, might have been there. Throughout the book, the subject matter has been dealt in such a simple, logical and systematic way that students will not find any difficulty to understand it. I express my thanks to my parents and all family members for their support. I would like to express my thanks to my friends. I am thankful to Asian Books Pvt. Ltd., New Delhi and its Production Team, especially Mr. A. S. Khan. Manisha Singhwal

7 Syllabus MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE MCA: CA-102 UNIT-I Set Theory: Definition of sets, countable and uncountable sets, Venn Diagrams, proofs of some general identities on sets. Relation: Definition, types of relation, composition of relations, Pictorial representation of relation, equivalence relation, partial ordering relation. Function: Definition, type of functions, one to one, into and onto function, inverse function, composition of functions, recursively defined functions. Mathematical Induction: Piano s axioms, Mathematical Induction Discrete Numeric Functions and Generating functions Simple Recurrence relation with constant coefficients, Linear recurrence relation without constant coefficients. UNIT-II Algebraic Structures: Properties, Semi group, Monoid, Group, Abelian group, properties of group, Subgroup, Cyclic group, Cosets, Permutation groups, Homomorphism, Isomorphism and Automorphism of groups Propositional Logic: Preposition, First order logic, Basic logical operations, Tautologies, Contradictions, Algebra of Proposition, Logical implication, Logical equivalence, Normal forms, Inference Theory, Predicates and quantifiers, Posets, Hasse Diagram and Lattices: Introduction, ordered set, Hasse diagram of partially, ordered set, isomorphic ordered set, well ordered set, properties of Lattices, and complemented lattices. UNIT-III Graphs: Simple graph, multi graph, representation of graphs, Bipartite, Regular, Planar and connected graphs, Euler graphs, Hamiltonian path and circuits, Graph coloring, chromatic number, isomorphism and Homomorphism of graphs. Tree: Definition, Rooted tree, properties of trees, binary search tree, tree traversal. UNIT-IV Theory of computation: Introduction, Alphabets, Strings and Languages, Kleene Closure, NFA, DFA, Conversion of NFA to DFA, Optimizing DFA FA with output: Moore machine, Mealy machine, Conversions. Regular expression (RE), Definition, Regular expression to FA, Arden Theorem, DFA to Regular expression, Non Regular Languages, Pumping Lemma for regular Languages. Application of Pumping Lemma, Closure properties of Regular Languages. UNIT-V Chomsky Hierarchy of language, Context-free grammar (CFG), Pushdown Automata (PDA), equivalence of PDA s and CFG s, Introduction Turing Machine (TM), construction of TM for simple problems. TM as Computer of Integer functions, Universal TM, Recursive and recursively enumerable languages, Halting problem, Introduction to Undecidability, Undecidable problems about TMs.

8 Contents UNIT I 1. Set Theory Relation Function Mathematical Induction UNIT II 1. Algebraic Structure Ring and Fields Propositional Logic Boolean Algebra Posets, Hasse Diagram and Lattices UNIT III 1. Graph Theory UNIT IV 1. Theory of Computation The Theory of Automata Regular Expression Formal Languages UNIT V 1. Context Free Grammar Pushdown Automata Turing Machine

9 Chapter 1 Chapter 2 Chapter 3 Chapter 4 Unit I Set Theory Relation Function Mathematical Induction 1

10 Set Theory CHAPTER The Theory of sets was developed by German Mathematician George Canter ( ) who defined a set as a collection aggregate of definite and distinguishable objects selected by means of some rules. 1.1 SETS-DEFINITION A set is well defined collection of objects or elements or numbers. There objects may be name of student points in Coordinate Geometry, letters of alphabets, numbers etc. Example of Sets: The collection of vowels in the English alphabets A = {a, i, e, o, u} The collection of All Natural numbers. N = {1, 2, 3,...} The collection of all even numbers B = {2, 4, 6, 8...} 1.2 REPRESENTATION OF SETS We denote set by capital letters like A, B, C... etc and the elements of sets are represented by small letters a, b, c.... If a is an element of set A then we can write a A i.e. a belongs to A or a is an element of A, and If a is not an element of A then we can write a A i.e. A does not belongs to A. There are two type of representation of sets Tabular or roster form Set builder or property or rule method Tabular or Roster Method In this form, all the elements of set are listed with in {} and elements being separated by commas. 3

11 4 Mathematical Foundation of Computer Science Example: The set of all natural numbers N = {1, 2, 3,...} Set of vowels V = {a, i, e, o, u} Question: Write the following sets in the roster form (i) A set of all factor of 12 Ans: A = {1, 2, 3, 4, 6, 12} (ii) B = set of all even integers less than 20 Ans: B = {2, 4, 6, 8, 10, 12, 14, 16, 18} Rule or Property or Set Builder Method In this form, a set is identify by its property i.e. the set of all those x such that each x has properties say. Example: A = {x : x is even no.} B = set of all natural number less than 8 B = {x : x N and x < 8} Question: Write the set E =,,,,... in set builder form n Ans: E = x: x= where nis natural number n +1 Note: 1. Order of elements are not important. It may be {1, 2, 3} or {1, 3, 2} both are same. 2. The repeated letters are taken only once each, like A = {1, 2, 3, 4} B = {1, 2, 3, 3, 4} both are same. 1.3 TYPES OF SETS Types of sets are following: Empty Sets A set containg no elements at all is called empty set or null set or void set. It is denoted by the symbol and {}. Example: N = {x : x N and 2 < x < 3} Ans: N = Example: A = {x : x 2 = 9 where x is even number} Ans: A =

12 Set Theory Singleton Set A set containing exactly one element is called singleton set. Example: The capital of India Ans: A = {Delhi} Example: A = {x : x N and x 2 9 = 0} Ans: A = {3} Note: { } is also an example of singleton set Finite Set A set is said to be finite set if it contains finite number of elements. Example: Set of all students in a particular class. Set of rivers Set of vowels Infinite Set A set which contains infinite number of elements is known as infinite set. Example: Set of all natural numbers Set of all prime numbers Set of all stars in the sky. 1.4 SUBSET A set with in set is called subset. If all the elements of a set A are also the elements of the set B, then set A is subset of B. Example: A = {a, b, c} B = {a, b, c, d} through this example it is clear that all the elements of A are also the elements of B. Therefore A is a subset of B it is denoted by A B if x A and x B in other hand if A is not subset of B i.e atleast one element of A does not belongs to B. Properties of Subset Theorem 1: Every set A is a subset of itself. i.e. A A Proof: Let A be any set. Then each element of A is clearly in A itself. Hence A A. Theorem 2: Null set is a subset of every set. Proof: Let A be any given set, then we have to Prove that A Let us suppose A i.e. is not subset of A if A, then there is atleast one element x in A.

13 6 Mathematical Foundation of Computer Science Such that x...(1) if x A then x {since is null set}...(2) From Equation (1) and (2) x and x, which is contradiction so our assumption is wrong, then A Proved Theorem 3: If A is subset of B and B is a subset of C then A is a subset of C. In symbols, A B and B C then A C. Proof: Let us suppose x be an element of A i.e. x A given that A B [ If x is an element of A then x is also element of B by the definition of subset] x A x B...(1) Similarly B C x B x C...(2) From equation (1) and (2) x A x C Therefore A C Proved 1.5 PROPER SUBSET Any subset A is said to be proper subset of another set B if A is subset of B but there exist atleast one element of B which is not belong to A. i.e. A B but A B we can write A B, A is a proper subset of B Example: A = {a, b} B = {a, b, c} every element of B is an element of set B but A B, then A is a proper subset of B. 1.6 SUPER SET If A is a subset of B then B is a superset of A i.e. B A 1.7 POWER SET The set of all subset of a given set A is called power set of A, denoted by P(A). Then total number of subset in a set can be defined by = 2 n where n is the number of element in a set. Example 1: If A = {1, 2} then P(A) = {, {1}, {2}, {1, 2}} Note: P(A) = {{ }, {1}, {2}, {1, 2} is wrong way of power set because { } is subset where is an element. Example: If A = {a, b, c} then power set P(A) = {, {a}, {b}, {c}, {a, b}, {b, c} {c, a}, {a, b, c}}

14 Set Theory 7 Theorem: The total number of subset of a finite set containg n element is 2 n Proof: Let A be a finite set containing n element Let 0 r n Consider those subset of A that has r element each. We know that number of ways in which r elements can be chossen out of n element is n C r. Therefore the number of subsets of A having r element each is n C r Hence, the total number of subset of A is = n C 0 + n C 1 + n C n C r +... = (1 + 1) n = 2 n Proved Question: Two finite sets have m and n elements. The total number of subset of the first set is 56 more than total number of subsets of second set. Find the value of m and n? Ans: We know that if a set has m elements then total number of subset of set = 2 m So, In first set total number of subset = 2 m In second set total number of subset = 2 n According to the question 2 m = 2 n m 2 n = 56 2 n (2 m n 1) = n = 8 2 n = 2 3 n = 3 Now, 2 m n 1 = 7 2 m n 1 = m n = 8 2 m n = 2 3 Taking power m n = 3 Put n = 3 m 3 = 3 m = 6 Hence the value of m is 6 and n is EQUAL SET Two sets A and B are equal set if they have exactly the same elements i.e. A = B if x A and x B Example: A = {1, 2, 3} B = {2, 3, 1} C = {1, 3, 2} A, B, C are equal sets. 1.9 CARDINAL NUMBER OF SETS The total number of elements in a set A is called cardinal number of given set. It is denoted by n(a). Example: A = {1, 2, 3} n(a) = 3

15 8 Mathematical Foundation of Computer Science Equivalent Set Two set A and B are equivalent set if their cardinal number are same for example. Let A = {1, 2, 3} B = {a, b, c} A and B both are equivalent because n(a) = 3 and n(b) = 3. Universal Set In any application of set theory, all the sets under investigation are likely to be considered as subset of particular set. The set is called universal set and denoted by U or universe of discourse. Example: Set of All Natural number of is the universe form in which we can choosen even number and odd number as a set. Operation on Sets In this section we will discuss several operation that will create a new set by two sets. 1. Union: The union of two set A and B denoted by A B is the set of all those elements which belongs to A or to B. A B = {x : x A or x B} Example: Let A = {1, 2} B = {1, 2, 3} A B = {1, 2} {1, 2, 3} = {1, 2, 3} 2. Intersection: The intersection of two set A and B denoted by A B is the set of all those elements which belongs to A and B both. A B = {x : x A and x B} Example: Let A = {1, 2, 3} B = {1, 2} A B = {1, 2, 3} {1, 2} = {1, 2} 3. Disjoint Set: If two set A and B do not have any element in common is called disjoint set i.e. A B =. Example: A = {1, 2} B = {4, 3} A B = i.e. A and B are disjoint set 4. Complement of a Set: Let U be the universal set and A by any subset of U. The absolute complement or simply complement of set A is denoted by A or A in other words, means A set of element which belongs to U but not belong to A. Example: Let U = {1, 2, 3, 4, 5, 6} A = {1, 2, 3} A = U A = {1, 2, 3, 4, 5, 6} {1, 2, 3} = {4, 5, 6} 5. Difference of Set: Two set A and B is called difference of A and B, denoted by A/B or A B is the set of elements which belongs to A but which do not belong to B A B = {x : x A but x B} Example: Let A = {1, 2, 3} B = {3, 4, 5} A B = {1, 2}

16 Mathematical Foundation of Computer Science Publisher : ASIAN ISBN : Author : Manisha Singhwal Type the URL : 34 Get this ebook

BASIC MATHEMATICAL TECHNIQUES

BASIC MATHEMATICAL TECHNIQUES CHAPTER 1 ASIC MATHEMATICAL TECHNIQUES 1.1 Introduction To understand automata theory, one must have a strong foundation about discrete mathematics. Discrete mathematics is a branch of mathematics dealing