Arithmetic and counting

This course aims to introduce the basic concepts of arithmetic and counting that are useful for beginning a bachelor's degree in mathematics.

Basic counting

• Cardinality of a finite set. Cardinality and set operations. Cardinality and injective, surjective, and bijective functions. Cardinality of a set of functions. Number of parts of a set. Indicator function.
• Introduction to infinite cardinals. Bijection between sets. Countability. Cantor's diagonal argument. X and P(X) do not have the same cardinality. R is uncountable.
• Arrangements, permutations, combinations (binomial coefficients), Pascal's triangle, binomial formula.
• General sieve formula (application to counting disturbances, surjections, etc.).
• Binary relation on a set. Equivalence relation, partition into equivalence classes, quotient of a set by an equivalence relation (examples on already known sets). Order relation, partial, total, examples.
• Applications to examples of elementary finite probabilities (number of favorable cases/total number of cases)

Elementary arithmetic in Z

• Integers, writing in a base.
• Divisibility, prime numbers (infinity, sieve algorithm). Euclidean division (Euclid's algorithm).
• GCD and LCM. Bézout's theorem (and extended Euclidean algorithm), relatively prime numbers, Euclid's lemma, Gauss's lemma. Diophantine equations ax + by = c. Prime factorization. Application: for n ∈ N, is either an integer or irrational.
• Modular arithmetic (congruences). Fermat's little theorem. Chinese remainder theorem.
• Study ofZ/nZ, viewed as a ring. Inversible,Z/nZ is a field if and only if n is prime. Reinterpretation of Bézout's theorem. Reinterpretation of Fermat's little theorem (definition of Euler's indicator, Euler's theorem). Reinterpretation of the Chinese remainder theorem.
• Illustration using cryptography.

S1 mathematics program (mainly reasoning and set theory) and high school mathematics programs (at least first-year mathematics specialization)

Recommended prerequisites:

S1 mathematics program (mainly reasoning and set theory) and high school mathematics programs (ideally mathematics specialization in the final year, or even advanced mathematics option).

Hourly volumes:

            CM: 30 hours

            Tutorial: 30 hours

            TP: 0

            Land: 0