MSC Semester I - Discrete Mathematics

Unit – I
Mathematical Logic
Propositional logic, propositional equivalences, predicates & quantifiers, rules of inference, direct proofs, proof by contraposition, proof by contradiction.
Boolean Algebra
Boolean functions and their representation, logic gates, minimization of circuits using Boolean identities and K-map.
Unit – II
Basic Structures
Set representations, set operations, functions, sequences and summations.
Division algorithm, modular arithmetic, solving congruences, applications of congruences.
Recursion
Proof by mathematical induction, recursive definitions, structural induction, generalized induction, recursive algorithms.
Unit – III
Counting
Basic counting principle, inclusion–exclusion for two sets, pigeonhole principle, permutations and combinations, binomial coefficients and identities, generalized permutations and combinations.
Recurrence Relations
Introduction, solving linear recurrence relations, generating functions, principle of inclusion–exclusion, applications of inclusion–exclusion.
Relations
Relations and their properties, representation of relations, closures of relations, equivalence relations, partial orderings.
Unit – IV
Graphs
Graph definitions, graph terminology, types of graphs, representation of graphs, graph isomorphism, connectivity of graphs, Euler and Hamilton paths and circuits, Dijkstra’s algorithm for shortest path, planar graphs – Euler’s formula and applications, graph coloring and applications.
Trees
Tree definitions, properties of trees, applications of trees – BST, Huffman coding, tree traversals: pre-order, in-order, post-order, prefix, infix, postfix notations, spanning trees – DFS, BFS, Prim’s and Kruskal’s algorithms.