FOUNDATION OF MATHEMATICS (MDC5001C)

UNIT-01: INTRODUCTION TO COMBINATORICS & ITS APPLICATION (20%)

• Counting Principles – Addition and Multiplication Rules
• Permutations (With and Without Repetition)
• Combinations and Related Identities
• Binomial Theorem and Pascal's Triangle
• Pigeonhole Principle (Basic and Generalized)
• Inclusion Exclusion Principle
• Derangements and Applications
• Counting with Repetition and Multisets
• Integer Partitions and Compositions
• Recurrence Relations and Their Solutions
• Generating Functions and Applications

UNIT-02: NUMERICAL METHODS (34%)

• Types of Errors: Absolute, Relative, Percentage
• Solutions of Algebraic and Transcendental Equations – Bisection Method, Regula-Falsi Method, Newton Raphson Method, Secant Method
• Solution of Linear System of Equations – Gauss Elimination, Gauss-Jordan, Gauss-Seidel and Jacobi Iterative Methods
• Interpolation – Newton's Forward and Backward Interpolation
• Lagrange's Interpolation; Newton's Divided Difference Interpolation

UNIT-03: OPERATIONS RESEARCH & LINEAR PROGRAMMING (24%)

• Basic concepts, scope and applications of OR
• Graphical solution for two-variable LPP
• Feasible region, optimal solution, multiple/unbounded/infeasible cases
• Standard form of LPP
• Introduction to Simplex algorithm and Tableau method
• Concept of Duality, formulation of Dual Problem Transportation Problem (TP) – IBFS using North-West Corner Rule and Least Cost Method; Optimality test using MODI method
• Degeneracy in TP
• Assignment Problem (AP) – Hungarian Method
• Introduction to Game Theory: Two-person zero-sum games, saddle point, pure and mixed strategies, dominance principle, graphical method
• Real-life applications

UNIT-04: LATTICES & BOOLEAN ALGEBRA (22%)

• Relations and ordering, Partially Ordered Sets (POSETs)
• Lattices as POSETs – properties, sub-lattices, direct product, homomorphism
• Complete lattices, bounds, distributive lattice, complemented lattices
• Introduction and properties of Boolean Algebra
• Sub Boolean algebra, join-irreducible, meet-irreducible, atoms, anti-atoms
• Stone's Representation Theorem (without proof)
• Applications