Year 2 Syllabus

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Discrete Mathematics +

Unit 1: Sets, Logic & Relations +
- Set Theory:
  - Sets, Subsets, Powerset
  - Set operations
  - Principle of inclusion-exclusion
  - Pigeonhole principle
- Logic Systems:
  - Propositional logic
  - Rules of inference
  - Predicate logic & quantifiers
  - Normal forms
- Proof Techniques:
  - Direct proof
  - Proof by contraposition
  - Proof by contradiction
- Combinatorics:
  - Permutations & combinations
  - Well-ordering principle
  - Mathematical induction
  - Complete induction
- Relations:
  - Binary relation properties
  - Equivalence relations & classes
  - Closures (reflexive, symmetric, transitive)
Unit 2: Functions & Boolean Algebra +
- Function Theory:
  - Function properties
  - Growth of functions
  - Permutation functions
- Order Structures:
  - Posets (Partially ordered sets)
  - Lattices
- Boolean Algebra:
  - Boolean expressions
  - Minimization techniques
- Number Theory:
  - GCD, LCM
  - Prime numbers
- Recurrence Relations:
  - Solution methods for linear recurrence
  - Generating functions
  - Divide-and-conquer recurrences
  - Master's theorem (with proof)
Unit 3: Group Theory +
- Algebraic Structures:
  - Semi-groups
  - Monoids
  - Groups (definition & properties)
- Group Properties:
  - Identity & inverses (uniqueness)
  - Subgroups
  - Cosets
  - Lagrange's theorem
- Morphisms:
  - Isomorphism
  - Homomorphism
- Advanced Concepts:
  - Permutation groups
  - Cayley's theorem (statement)
  - Normal subgroups
  - Quotient groups
  - Coding applications
Unit 4: Graph Theory +
- Fundamentals:
  - Graph terminology
  - Planar graphs
  - Euler's formula (proof)
- Special Graphs:
  - Eulerian paths/circuits
  - Hamiltonian paths/circuits
- Graph Coloring:
  - Chromatic number
  - Five color theorem (proof)
- Algorithms:
  - Shortest path algorithms
  - Minimum spanning trees
  - DFS & BFS implementations
  - Tree constructions
- Complexity Analysis:
  - Connected components
  - MST algorithm analysis

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