Set Operations: Union, intersection, and complement of sets
Venn diagrams
Examples of set operations
Subsets and Supersets: Definition of subsets and supersets
Proper subsets
Subset relationships
Cartesian Products and Relations: Cartesian product of sets
Ordered pairs
Relations between sets
Functions: Definition of functions
Domain and codomain
Types of functions (injective, surjective, bijective)
Set Identities and Laws: Associative, commutative, and distributive laws, De Morgan's laws, Idempotent laws
Cardinality and Counting: Cardinality of sets, Finite and infinite sets, Countable and uncountable sets
Applications of Set Theory: Set theory in computer science, Set theory in probability and statistics
Real-life applications: Set theory in computer science, Set theory in probability and statistics, Real-life applications
Unit-02 Matrices and Determinant (25%)
Definition of a matrix
Row matrix
Column matrix
Square matrix
Diagonal matrix
Scalar matrix
Identity matrix
Zero matrix
Symmetric matrix
Skew-symmetric matrix
Introduction to Determinants
Invertible matrix: Definition of a determinant
Properties of determinants: Linearity , Multiplicativity , Transposition,
Special properties of determinants of triangular matrices
Invertible matrix: Definition, Properties of invertible matrices
Relationship between invertibility and the determinant
Computation of Inverse using Definition: Method of finding the inverse using adjoint and determinant
Properties of the inverse matrix
Examples of computing inverses for 2x2 and 3x3 matrices
Conditions for the existence of inverses
Simultaneous Solution of Set of Linear Equations using Cramer ’ s Rule
System of linear equations: Definition, representing linear equations in matrix form
Cramer's Rule for solving systems of linear equations: Statement of Cramer's Rule, Conditions for the applicability of Cramer's Rule, Advantages and limitations of Cramer's Rule
Application of Matrix in Various Fields and Everyday Life: Real-life examples demonstrating the use of matrices in everyday scenarios
Graph Theory and Group Theory (28%)
Introduction to Basic Graph Terminology: Definition of a graph, Nodes (vertices) and edges
Directed and undirected graphs
Adjacency matrix and adjacency list representation
Degree of a vertex, Path, cycle, and connected graphs
Study of Various Types of Graphs: Tree and forest graphs, Bipartite graphs, Complete graphs, Planar and non-planar graphs, Weighted graphs, Directed acyclic graphs (DAGs)
Euler’s Paths and Circuits
Hamiltonian Paths: Eulerian paths and circuits
Euler's theorem and criteria for existence
Hamiltonian paths and cycles
Dirac's theorem and sufficient conditions for Hamiltonian cycles
Application of Graphs: Network analysis and optimization, Transportation and logistics, Social network analysis, Computer science
Introduction to Groups, Subgroups & Semigroups: Definition of groups and their properties
Subgroups and subgroup criteria
Cyclic groups and generator elements
Semigroups and their operations
Products and Quotients of Groups: Group product and direct product
Quotient groups and homomorphisms
Isomorphism theorems for groups
Cosets and Lagrange's theorem
Applications of Group & Graph Theory: Cryptography and coding theory, Symmetry and crystallography , Graph theory in computer science, Combinatorial optimization and game theory
Unit-04 Number Theory (22%)
Divisibility: Understanding the concept of divisibility , prime numbers, composite numbers, and the fundamental theorem of arithmetic
Prime Numbers: Properties of prime numbers, prime factorization, prime counting function, distribution of prime numbers, and prime number theorems
Congruences: Modular arithmetic, congruence relations, solving linear congruences
Chinese remainder theorem, and applications in cryptography
Arithmetic Functions: Euler's totient function, Möbius function, divisor functions, and their properties
Diophantine Equations: Equations involving integer solutions, linear Diophantine equations
Fermat's Last Theorem, and applications in cryptography and number theory
Quadratic Residues: Quadratic residues and non-residues, quadratic reciprocity theorem, and applications in cryptography and number theory
Arithmetic Progressions: Understanding arithmetic progressions, arithmetic functions in arithmetic progressions, and Dirichlet's theorem on primes in arithmetic progressions
Primitive Roots and Indices: Primitive roots modulo n, index of an integer modulo n, and their properties
Pell's Equation: Solving Pell's equation, properties of solutions, and applications in number theory and algebra
Transcendental Numbers: Introduction to transcendental numbers
Lindemann-Weierstrass theorem, and properties of transcendental numbers.