# Discrete Mathematics I

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Set Theory:Introduction to the theory of sets; combination of sets; power sets; finite and infinite sets; principle of inclusion and exclusion; selected problems from each topic;Logic:Proposition, predicate logic, logic operators, logic proposition and proof, method of proofs – Mathematical Induction Different forms of the principle of mathematical induction. selected problems on mathematical induction – Discrete Probability:Counting principles. Random experiment; sample space; events; axioms of probability; conditional probability. Theorem of total probability; Bayes’ theorem. Application to information theory: information and mutual information;Graph theory:Path, cycles, handshaking theorem, bipartite graphs, sub-graphs, graph isomorphism, operations on graphs, Eulerian graphs and Hamiltonian graphs, planar graphs, Euler formula, traveling salesman problem, shortest path algorithms;Relations:Definitions and properties; Equivalence relations and equivalence classes. Representations of relations by binary matrices and digraphs; operations on relations. Closure of a relations; reflexive, symmetric and transitive closures. Warshall’s algorithm to compute transitive closure of a relation;Partially Ordered Sets and Lattices – Partial order relations; POSETS; lattices – Boolean Algebra and Boolean Functions Introduction to Boolean algebra and Boolean functions. Different representations of Boolean functions. Application of Boolean functions to synthesis of circuits – Discrete Numeric Functions:Introduction of discrete numeric functions; asymptotic behaviour; generating functions;Recurrence Relations:Linear recurrence relations with constant coefficients (homogeneous case); discussion of all the three sub-cases. Linear recurrence relations with constant coefficients (non-homogeneous case); discussion of several special cases to obtain particular solutions. Solution of linear recurrence relations using generating functions

### Course Curriculum

 Introduction to the theory of sets Details 56:9 Set operation and laws of set operation Details 49:8 The principle of inclusion and exclusion Details 47:29 Application of the principle of inclusion and exclusion Details 55:28 Fundamentals of logic Details 46:3 Logical Inferences Details 43:6 Methods of proof of an implication Details 50:27 First order logic(1) Details 42:34 First order logic(2) Details 43:17 Rules of influence for quantified propositions Details 38:22 Mathematical Induction(1) Details 43:21 Mathematical Induction(2) Details 52:38 Sample space, events Details 57:57 Probability, conditional probability Details 57:57 Independent events, Bayes theorem Details 42:48 Information and mutual information Details 53:46 Basic definition Details 41:36 Isomorphism and sub graphs Details 44:1 Walks, paths and circuits operations on graphs Details 54:48 Euler graphs, Hamiltonian circuits Details 42:9 Shortest path problem Details 45:42 Planar graphs Details 41:34 Basic definition. Details 1:5:9 Properties of relations Details 47:30 Graph of relations Details 49:33 Matrix of relation Details 51:34 Closure of relaton(1) Details 57:54 Closure of relaton(2) Details 56:52 Warshall’s algorithm Details 1:3:17 Partially ordered relation Details 49:21 Partially ordered sets Details 55:6 Lattices Details 53:57 Boolean algebra Details 58:42 Boolean function(1) Details 1:2:51 Boolean function(2) Details 57:36 Discrete numeric function Details 1:36 Generating function Details 59:26 Introduction to recurrence relations Details 49:7 Second order recurrence relation with constant coefficients(1) Details 50:58 Second order recurrence relation with constant coefficients(2) Details 59:35 Application of recurrence relation Details 55:59

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