**This workshop will aim to touch upon some of these topics **

**Foundation & Algebra**

- Foundation: Basic concepts of set, union, intersection, etc., lots of examples to be given. Index set, Cartesian product of sets, relations, equivalence relations, functions, 1-1 function, onto function, Partitions
- Natural numbers, Well-ordering principle, induction principle, weak induction principle and their equivalence; Divisibility, Division algorithm; GCD and LCM; Bezoutâ€™s identity. irreducible elements and prime elements; Fundamental theorem of arithmetic
- Linear Diophantine equations. Congruences; arithmetic of congruences; Chinese remainder theorem
- Composition of functions and permutations, Decomposition
- Theory of Groups, Isomorphism, Cosets and Langranges theorem, Cyclic groups, Cayley's theorem, Group action

**Analysis**

- Upper and lower bounds, LUB and GLB, LUB axiom
- Archimedean property, density of rationals and irrational, nested interval theorem
- Sequences in R and convergence: algebra of convergent sequences, monotone sequences, Cauchy sequences and their convergence, subsequences, Bolzano- Weierstrass theorem, recursively defined sequences
- Continuity: sequential definition, epsilon-delta definition and their equivalence. Intermediate value theorem

** Linear Algebra**

- Systems of linear equations, Gaussian elimination, Geometric interpretation of solution sets, non-homogeneous and homogeneous systems of linear equations. Existence of nontrivial solutions for a homogeneous system of m linear equations in n variables with m < n
- Vector spaces, vector subspaces, vector subspaces generated by a set of vectors, Linear dependence and independence, basis, dimension formula for the sum a two vector subspaces
- Linear Transformations, Kernel and image, Rank-nullity theorem, Matrix Representation w.r.t ordered bases