For each of the following teachings, I have participated, in team, to the creation of worksheet and tests, and grading exam papers.
A first part on logic: Propositional calculus, truth tables, logical connectives, and quantification. Basic set-theoretical structures: Maps, one-to-one functions, surjective functions, bijections, equivalence relations, order relations. Many examples of induction arguments.
A second part on arithmetic: Greatest common divisor, Euclidean algorithm, Bézout's identity, Fermat's little theorem, simple Diophantine equations, Euler's theorem, Chinese remainder theorem, RSA.
Students were also asked to implement several basic algorithms.
A first part on polynomial and generalizations: Cardano's method, multivariate polynomials, symmetric polynomials, elementary symmetric polynomials. Formal series and relations with combinatorics.
A second part on real Hilbert spaces: Bilinear forms, dot product, orthogonality, orthonormal bases, Gram-Schmidt process, orthogonal sub-spaces.
Students, using Openoffice calc, apply their knowledge of statistics to study a set of data.
Complex numbers: Roots of unity, resolution of equations in C, fundamental theorem of algebra.
Polynomials: Division, gcd, roots and multiplicity, irreducible polynomials in R and C, rational functions, and partial fraction decomposition.
Linear algebra: Gaussian elimination and introduction to matrices.
Vector spaces, subspaces, sum of subspaces, direct sum. Spanning set, linearly independent vectors, bases, dimension. Linear maps, kernel, rank, image, rank-nullity theorem, matrix of a linear map, operations on matrices, determinant, multilinear forms, alternating forms.
Real numbers: ordering, set-theoretical operations on intervals, least upper bound. Study of integer sequences, notion of limit, convergent sequences, operations on convergent sequences, monotonic sequences, Cauchy sequences.
Functions: one-to-one functions, surjective functions, bijections. Limits, continuous functions, characterization of continuous functions, intermediate value theorem, monotonous continuous functions.
Differentiable functions, Rolle's theorem, mean value theorem, Taylor's theorem and Taylor expansion. Inverse functions of classical functions.
Computation of primitives and integrals: Integration by parts, integration by substitution, integrals of rational functions. First-order partial differential equation.
Bilinear forms, signature of a quadratic form, computation of maxima and minima of a function with several variables.
The goal is to learn how to write a mathematic proof, the use of classical arguments (induction, reductio ad absurdum, contraposition, etc.), the use of logical connectives and quantification.
Many example of formal definitions are studied, set-theoretic operations, maps, one-to-one functions, surjective functions, bijections, binary relations, equivalence relations, partial order, operations, groups, rings, and fields.
A first part is on linear algebra: resolution of linear systems, finite dimensional vector spaces, linear maps, kernel, image.
A second part on computation of primitive and integrals: Integration by parts, integration by substitution, integrals of rational functions, linear differential equations.