syllabus/maths.md

UNIT I: Mathematical Logic, Sets, Relations, and Functions

Mathematical Logic:

• Notations, Algebra of Propositions & Propositional functions
• Logical connectives, Truth values & Truth tables
• Tautologies & Contradictions, Normal Forms
• Predicate Calculus, Quantifiers

Set Theory:

• Sets, Subsets, Power sets, Complement, Union and Intersection
• De Morgan's Law, Cardinality

Relations:

• Cartesian Products, relational Matrices, properties of relations, equivalence relations

Functions:

• Injection, Surjection, Bijection, Composition of Functions, Permutations, Cardinality
• Characteristic functions, Recursive definitions, Finite induction

UNIT II: Lattices & Boolean Algebra

Lattices:

• Lattices as Algebraic Systems, Sublattices
• Some special lattices: Complement, Distributive, Modular

Boolean Algebra:

• Axiomatic definitions of Boolean algebra as algebraic structures with two operations
• Switching Circuits

UNIT III: Groups, Fields, & Rings

Groups:

• Definition of groups, axioms, permutation groups
• Subgroups, co-sets, normal subgroups, free subgroups
• Grammars, language

Fields & Rings:

• Definition and structure of fields and rings
• Minimal Polynomials, Irreducible Polynomials
• Polynomial roots & its Applications

UNIT IV: Graphs

Graphs:

• Simple Graph, Multigraph & Pseudograph
• Degree of a Vertex, Types of Graphs, Subgraphs, Isomorphic Graphs
• Operations on Graphs, Paths, Cycles, and Connectivity
• Euler and Hamilton Graphs, Shortest Path Problems (BFS, Dijkstra's Algorithm)
• Representation of Graphs, Planar Graphs, Applications of Graph Theory

UNIT V: Trees

Trees:

• Definition and properties of trees, pendant vertices in a tree, center of a tree
• Spanning tree, Binary tree, Tree traversal
• Applications of trees in computer science