This course introduces the fundamentals of mathematical proof and problem-solving, facilitating the transition from basic methods to more advanced mathematical concepts and reasoning.

Lectures-Tutorials-Practicals-Credits: 4-0-0-4

Week 1: Foundations of mathematics; Sequences and summations (sigma notation); Set theory; Cardinality and bijections; Natural numbers and integers.

Week 2: Infinite sets and real numbers; Cantor's diagonalization; Ordered fields and completeness; Least upper bound property; Mathematical logic.

Week 3: Number theory basics; Divisibility and GCD; Euclidean algorithm and its proof.

Week 4: Prime numbers and modular arithmetic; Fermat's Little Theorem; Fundamental Theorem of Arithmetic; Inclusion-exclusion principle.

Week 5: Combinatorial principles; Pigeonhole principle; Sieve of Eratosthenes; Distribution and gaps of primes.

Week 6: Combinatorics fundamentals; Binomial coefficients and theorem; Permutations; Lattice paths; Random sampling.

Week 7: Graph theory basics; Connectivity and distance; Trees and adjacency matrices; Graph coloring; Sperner's lemma.

Week 8: Limits and continuity; Properties of limits; Sequential continuity; Intermediate Value Theorem.

Week 9: Advanced limits; Fibonacci sequence behavior; Sandwich and shifting lemmas.

Week 10: Differential calculus; Derivatives; Higher-order derivatives (Faà di Bruno's formula).

Week 11: Integral calculus; Riemann integrals; Mean Value Theorem.

Week 12: Uniform continuity; Fundamental Theorem of Calculus.