# Discrete Mathematics for GTU 18 Course (IV- CSE/IT- 3140708) & Discrete Mathematics and Graph Theory GTU 20 Course (CSE(Artificial Intelligence & Machine Learning - 3144202)) (Decode)

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1. Set Theory : Basic Concepts of Set Theory: Definitions. lnclusion, Equality of Sets, Cartesian product, The Power Set, Some operations οη Sets, Venn Diagrams. Some Basic Set ldentities. Functions : Introduction & definition, Co-domain, range, image, value of a function: Examples, surjective, injective, bijective: examples; Composition of functions, examples; Inverse function, ldentity map, condition of a function to be invertible, examples: Inverse of composite functions. Properties of Composition of functions: Counting The Basics of Counting, The Pigeonhole Principle, Permutations and Combinations, Binomial Coefficients, Generalized Permutations and Combinations. Generating Permutations and Combinations (Chapter - ι. 2, 3) 2. Propositional Logic : Definition. Statements & Notation. Truth Values. Connectives, Statement Formulas & Truth Tables, Well-formed Formulas. Tautologies, Equivalence of Formulas, Duality Law, Tautological Implications, Examples Predicate Logic : Definition of Predicates: Statement functions. Variables. Quantifiers. Predicate Formulas, Free & Bound Variables: The Universe of Discourse, Examples, Valid Formulas & Equivalences, Examples (Chapter - 4, 5) 3. Relations : Definition, Binary Relation, Representation, Domain, Range, Universal Relation, Void Relation, Union, Intersection, and Complement Operations οη Relations, Properties of Binary Relations ίη a Set: Reflexive, Symmetric, Transitive, Anti-symmetric Relations. Relation Matrix and Graph of a Relation: Partition and Covering of a Set, Equivalence Relation, Equivalence Classes, Compatibility Relation, Maximum Compatibility Block, Composite Relation. Converse of a Relation. Transitive Closure of a Relation R ίη Set Χ (iv) Partial Ordering : Definition, Examples, Simple or Linear Orderinq, Totally Ordered Set (Chain). Frequently Used Partia!Iy Ordered Relations, Representation of Partially Ordered Sets, Hesse Diaqrams, Least & Greatest Members, Minimal & Maximal Members, Least Upper Bound (Supremum). Greatest Lower Bound (infimum). We!I-ordered Partia!Iy Ordered Sets (Posets). Lattice as Posets, complete, distributive modular and complemented Iattices Boolean and pseudo Boolean Iattices. (Definitions and simple examples only) Recurrence Relation : Introduction, Recursion, Recurrence Relation, Solvinq, Recurrence Relation. (Chapter - 6, 7, 8) 4. Algebraic Structures : Alqebraic structures with one binary operation­ Semiqroup, Monoid, Group, Subqroup, normal subqroup, qroup Permutations. Coset, homomorphic subqroups, Laqranqe's theorem. Conqruence relation and quotient structures. Alqebraic structures (Definitions and simple examples only) with two binary operation- Rinq, Inteqral domain and field. (Chapter - 9) 5. Graphs : Introduction, definition, examples; Nodes, edqes, adjacent nodes. directed and undirected edqe, Directed qraph, undirected qraph, examples; lnitiatinq and terminatinq nodes, Loop (slinq), Distinct edqes, Para!Iel edqes, Multi-qraph, simple qraph, weiqhted qraphs. examples, Isolated nodes, Nu!I qraph; Isomorphic qraphs, examples; Deqree, lndeqree, out-deqree. total deqree of a node, examples; Subqraphs, definition, examples; Converse (reversal or directional duaI) of a diqraph, examples; Path : Definition. Paths of a qiven qraph, lenqth of path. examples; Simple path (edqe simple). elementary path (node simple). examples; Cycle (circuit), elementary cycle, examples; Reachability : Definition. qeodesic, distance. examples; Properties of reachability, the trianqle inequality; Reachable set of a qiven node, examples, Node base, examples; Connectedness : Definition. weaRly connected, stronqly connected. unilatera!Iy connected. examples; Stronq, weaR, and unilateral components of a qraph, examples, Applications to represent Resource allocation status of an operatinq system, and detection and correction of deadlocRs; Matrix representation of qraph : Definition, Adjacency matrix, boolean (or bit) matrix, examples; Determine number of paths of Ienqth η throuqh Adjacency matrix, examples; Path (Reachability) matrix of a qraph, examples; Warsha!I's alqorithm to produce Path matrix, Flowchart. Trees : Definition, branch nodes, leaf (terminaI) nodes, root, examples: Different representations of a tree, examples; Binary tree, m-ary tree, Fu!I (or complete) binary tree, examples; Convertinq any m-ary tree to a binary tree. examples; Representation of a binary tree : LinRed-list; Tree traversal : Pre-order. in-order. post-order traversal, examples, alqorithms; Applications of List structures and qraphs. (Chapters - 10, 11)

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