Arithmetic and counting

The aim of this course is to introduce the elementary concepts of arithmetic and enumeration that will be useful at the beginning of a bachelor's degree in mathematics.

Elementary enumeration

• Cardinal of a finite set. Cardinal and set operations. Cardinal and injective, surjective and bijective applications. Cardinal of a set of applications. Number of parts of a set. Indicator function.
• Introduction to infinite cardinals. Bijection between sets. Denumerability. Cantor's diagonal argument. X and P(X) do not have the same cardinal. R is uncountable.
• Arrangements, permutations, combinations (binomial coefficients), Pascal's triangle, binomial formula.
• General screen formula (application to counting disturbances, overjections, 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 finite elementary probabilities (number of favorable cases/total number of cases)

Elementary arithmetic in Z

• Integers, writing in a base.
• Divisibility, prime numbers (infinitude, sieve algorithm). Euclidean division (Euclid's algorithm).
• PGCD and PPCM. Bézout's theorem (and extended Euclid's algorithm), prime numbers, Euclid's lemma, Gauss's lemma. Diophantine equations ax + by = c. Decomposition into product of primes. Application: for n ∈ N, is either an integer or irrational.
• Modular arithmetic (congruences). Fermat's little theorem. Chinese remainder theorem.
• Study of Z/nZ, seen as a ring. Inversibles, Z/nZ is a body if and only if n is prime. Reinterpretation of Bézout's theorem. Reinterpretation of Fermat's little theorem (definition of Euler's indicatrix, Euler's theorem). Reinterpretation of the Chinese remainder theorem.
• Illustrated by cryptography.

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

Recommended prerequisites :

S1 mathematics syllabus (mainly reasoning and set theory) and high school mathematics syllabus (ideally senior year mathematics specialization, or even expert mathematics option).

Hourly volumes* :

            CM: 30 h

            TD: 30 h

            TP: 0

            Land: 0