Module 1: Fundamentals of Logic: This includes basic logical connectives, truth tables, Laws of Logic, logical implication, Rules of Inference, quantifiers, definitions, and proof techniques such as Direct, Indirect, and Contradiction. Module 2: Properties of Integers & Principles of Counting: Topics covered are Mathematical Induction (including the Well-Ordering Principle and recursive definitions), rules of sum and product, permutations, combinations, the Binomial Theorem, and combinations with repetition. Module 3: Relations and Functions: This module includes Cartesian products, different types of functions (One-to-One, Onto), the Pigeon-hole Principle, properties of relations, Zero-One matrices, directed graphs, Partial Orders (Hasse Diagrams), equivalence relations, and partitions. Module 4: Principle of Inclusion-Exclusion & Recurrence Relations: Key topics are the Principle of Inclusion and Exclusion and its generalizations, Derangements, and Recurrence Relations, focusing on solving first-order and second-order linear homogeneous relations with constant coefficients. Module 5: Introduction to Group Theory: This module introduces definitions and examples of groups (like the Klein 4-group and additive/multiplicative groups), subgroups, cyclic groups, cosets, and Lagrange's Theorem.
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