Discrete Mathematics

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Yogesh Goyal and Kalu Ram Saini
University of Rajasthan - UOR, BCA, Part II
Syllabus
(BCA202) DISCRETE MATHEMATICS
 
Unit-I
Number Systems: Number systems – natural numbers, integers, rational numbers, real numbers, complex numbers, arithmetic modulo a positive integer. Radix r representation (decimal and binary), Change of radix (decimal to binary and vice versa).
Binomial Theorem and Mathematical Induction: Binomial theorem for positive integral indices, general and middle term in binomial expansion with simple applications. Some simple problems of Principle of Mathematical induction.
Recurrence Relations and Generating Functions: Recurrence relation, linear recurrence relation with constant coefficients, solution of linear recurrence relation with constant coefficients. Generating functions, Solution of recurrence relations using generating functions.
 
Unit-II
Sets: Definition of sets, representation of sets, type of sets, Operations on sets, Sub sets, Power set, Universal set, Complement of a set, Union and Intersection of two sets, Venn diagrams, De-Morgans law of sets, Partition of sets, Duality Principles.
Relations: Relation, Types of relations- reflexive, symmetric, anti-symmetric, transitive, equivalence and partial order relation. Relation and diagraphs, Cartesian product of two sets.
Functions: Function, domain and range, One to one and onto functions, composite functions, inverse of a functions. Binary operations.
 
Unit-III
Logic and Proofs: Proposition, Conjunction, Disjunction, Negation, Compound proposition, Conditional propositions (Hypothesis, conclusion, necessary and sufficient condition) and Logical equivalence, De Morgan’s law, Tautology and contradiction, quantifiers, universally quantified statements, component of a Mathematical system (axiom, definitions, undefined terms, theorem, lema and corollary), proofs (direct proofs, indirect proofs, proof by contra-positive), Mathematical Induction.
Boolean Algebra: Definition and Laws of Boolean Algebra, Boolean functions, Simplification of Boolean functions, Special forms of Boolean functions, Application of Boolean algebra (open and closed switches, switches in series and parallel). Logic gates and Circuits.
 
Unit-IV
Graph: Basic terminology, directed and undirected graphs, path and connectivity, types of graphs-Null, Regular, Complementary, Complete, Weighted and Bipartite. Subgraphs, Operation on graphs- union, intersection, complement, product and composition. Representation of graphs in computer memory (matrix representation)”. Adjacency matrix, Incidence matrix. Fusion of graphs. Isomorphic and Homeomorphic graphs, paths and cycles, Eulerian and Hamiltonian graphs, shortest path algorithm. Planar graphs, graph coloring. S Shortest path algorithms. Travelling salesman problem.
 
Unit-V
Tree: Definition of tree, Fundamental terminologies-Node, Child, Parent, Root, Leaf, Level, Height and Subling. Rooted trees, Ordered trees, Binary tree, Complete binary tree, Tree of an algebraic expression, Tree searching (traversal algorithms) – Preorder, Inorder and Postorder. Distance and centre, Relation between general tree and binary tree, Spanning trees, Algorithms for minimal spanning trees (Kruskal’s and Prim’s). Game tree.
UOR2017/BCA/2/03
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