Note for Discrete Mathematics - DMS By tarunesh singh
- Discrete Mathematics - DMS
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B. TECH. DISCRETE MATHEMATICS (I.T & Comp. Science Engg.) SYLLABUS B.Tech (CSE/IT, Discrete Mathematical Structures) Unit I Logic: Propositional equivalence, predicates and quantifiers, Methods of proofs, proof strategy, sequences and summation, mathematical induction, recursive definitions and structural induction, program correctness. Counting: The basics of counting, the pigeonhole principle, permutations and combinations, recurrence relations, solving recurrence relations, generating functions, inclusion-exclusion principle, application of inclusion-exclusion. Unit II Relations: Relations and their properties, n-array relations and their applications, representing relations, closure of relations, equivalence of relations, partial orderings. Graph theory: Introduction to graphs, graph terminology, representing graphs and graph isomorphism, connectivity, Euler and Hamilton paths, planar graphs, graph coloring, introduction to trees, application of trees. Unit III Group theory: Groups, subgroups, generators and evaluation of powers, cosets and Lagrange's theorem, permutation groups and Burnside's theorem, isomorphism, automorphisms, homomorphism and normal subgroups, rings, integral domains and fields. Unit IV Lattice theory: Lattices and algebras systems, principles of duality, basic properties of algebraic systems defined by lattices, distributive and complimented lattices, Boolean lattices and Boolean algebras, uniqueness of finite Boolean expressions, prepositional calculus. Coding theory: Coding of binary information and error detection, decoding and error correction. Text Books: 1) K.H. Rosen: Discrete Mathematics and its application, 5th edition, Tata McGraw Hill.Chapter 1(1.1-1.5), Chapter 3(3.1-3.4,3.6), Chapter 4(4.1-4.3,4.5), Chapter 6(6.1,6.2,6.4-6.6) Chapter 7(7.1-7.6), Chapter 8(8.1-8.5,8.7,8.8) 2. C. L. Liu: Elements of Discrete Mathematics, 2 nd edition, TMH 2000. Chapter 11(11.1 – 11.10 except 11.7), Chapter 12(12.1 – 12.8) 3.B.Kalman: Discrete Mathematical Structure, 3 rd edition, Chapter 11(11.1,11.2)
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Justification of Learning the Subject: What is Discrete Mathematics? Consider an analog clock (One with hands that continuously rotate and show time in continuous fashion) and a digital clock (It shows time in discrete fashion). The former one gives the idea of Continuous Mathematics whereas the later one gives the idea of Discrete Mathematics. Thus, Continuous Mathematics deals with continuous functions, differential and integral calculus etc. whereas discrete mathematics deals with mathematical topics in the sense that it analyzes data whose values are separated (such as integers: Number line has gaps) Example of continuous math – Given a fixed surface area, what are the dimensions of a cylinder that maximizes volume? Example of Discrete Math – Given a fixed set of characters, and a length, how many different passwords can you construct? How many edges in graph with n vertices? How many ways to choose a team of two people from a group of n? Why do you learn Discrete Mathematics? This course provides some of the mathematical foundations and skills that you need in your further study of Information Technology and Computer Science & Engineering. These topics include: Logic, Counting Methods, Relation and Function, Recurrence Relation and Generating Function, Introduction to Graph Theory And Group Theory, Lattice Theory and Boolean Algebra etc. .
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Unit I PROPOSITIONAL LOGIC AND COUNTING THEORY OBJECTIVES: After going through this unit, you will be able to : Define proposition & logical connectives. To use the laws of Logic. Describe the logical equivalence and implications. Define arguments & valid arguments. To study predicate and quantifier. Test the validity of argument using rules of logic. Give proof by truth tables. Give proof by mathematical Induction. Discuss Fundamental principle of counting. Discuss basic idea about permutation and combination. Define Pigeon hole principle. Study recurrence relation and generating function. INTRODUCTION : Mathematics is assumed to be an exact science. Every statement in Mathematics must be precise. Also there can’t be Mathematics without proofs and each proof needs proper reasoning. Proper reasoning involves logic. ‘Logic’ is the science of reasoning. The dictionary meaning of The rules of logic give precise meaning to mathematical statements. These rules are used to distinguish between valid & invalid mathematical arguments. In addition to its importance in mathematical reasoning, logic has numerous applications in computer science to verify the correctness of programs & to prove the theorems in natural & physical sciences to draw conclusion from experiments, in social sciences & in our daily lives to solve a multitude of problems.
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The area of logic that deals with propositions is called the propositional calculus or propositional logic. The mathematical approach to logic was first discussed by British mathematician George Boole; hence the mathematical logic is also called as Boolean logic. In this chapter we will discuss a few basic ideas. PROPOSITION (OR STATEMENT) A proposition (or a statement) is a declarative sentence that is either true or false, but not both. A proposition (or a statement) is a declarative sentence which is either true or false but not both. Imperative, exclamatory, interrogative or open sentences are not statements in logic. Example 1 : For Example consider, the following sentences. (i) VSSUT is at Burla. (ii) 2+3=5 (iii) The Sun rises in the east. (iv) (v) (vi) Do your home work. What are you doing? 2+4=8 (vii) (viii) (ix) ix) (x) 5<4 The square of 5 is 15. x 3 2 May God Bless you! All of them are propositions except (iv), (v),(ix) & (x) sentences ( i), (ii) are true, whereas (iii),(iv), (vii) & (viii) are false. Sentence (iv) is command, hence not a proposition. ( v ) is a question so not a statement. ( ix) is a declarative sentence but not a statement, since it is true or false depending on the value of x. (x) is a exclamatory sentence and so it is not a statement. Mathematical identities are considered to be statements. Statements whi ch are imperative, ex cl am at or y, i n t e r r o g a t i v e or open are not statements in logic.
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