Algebra and Geometry
No prerequisites are required.
The exam consists of a written exam based on the resolution of the exercises related to the subjects discussed in the course and on theoretical questions, followed by an optional interview in which, besides supplementing and correcting the written test exercises, the focus will be on the assessment of the acquisition and of the correct understanding of the content of the course.
Examples of past written tests can be found on the e-learning site of the course. The maximum rate that can be get at the written exam is 28. Students can
have the interview is their rate at the written exam is at least 25.
At the middle and at the end of the course, it is possible to support written tests, which, if passed both, are valid as a written part of the exam and allow access to the oral exam.
Knowledge and understanding skills
The course aims to provide basic knowledge of discrete mathematics, such as sets, functions, equivalence equations and algebraic structures, and linear algebra as linear and matrix systems and analytical geometry. In addition to being an integral part of the cultural baggage of a student of a science degree, these knowledge are designed to form the abstraction of problems and information through symbolic and mathematical representation. The course will complement the more theoretical and methodological aspects of mathematics, with the more technical aspects that allow for the resolution of exercises and which make mathematics an understanding and computing tool in various application areas.
Knowledge and understanding skills applied
During the course, emphasis will be placed on examples related to computer applications and especially algorithms. In particular, aspects relating to the understanding of properties of natural numbers such as recursion and induction are emphasized. An important part of the course will be devoted to exercises, always emphasizing that in order to be able to perform an exercise there is a need for total understanding of the subject being discussed.
Autonomy of judgment and communication skills
Expected learning outcomes include not only knowledge of terms and technical outcomes, but also the ability to master a mathematical argument by distinguishing premise and conclusions. In this perspective, the student's technical language needs to expand so that it can express abstract mathematical concepts.
Ability to learn
During the course, the importance of an appropriate study method will be emphasized, in particular by seeking to encourage a critical study (such as asking for some mathematical affirmations), in order to make students aware of their gaps.
Theorems and demonstration methods: implication, counter-nominal, demonstrations by absurdity. Quantifiers and denials. Induction principle, examples and exercises. (4h)
Sets, elements of a set, belonging and inclusion, subsets, set of parts of a set, cardinality of a finite set, Venn diagrams. Transactions between sets: union, intersection, and complement. Couples and Cartesian product. Counting elements of finite sets. (4h)
Relationships, binary relations, reflective, symmetrical and transitive properties. Equivalence reports, class of equivalence, quotient set. Partitions, the fundamental theorem of equivalence relations. Order reports, examples: divisibility between integers, prefixes, inclusions between sets. Non-comparable elements. Maximum and minimum. Lower bound and upper bound. (6h)
Functions, Domain and Codomain, Image and Pre-Image. Injective, surjective and bijective functions. Reverse of a bijective function. Composition of functions. (4h)
Combination Calculation Elements: Simple and Repetitive Layouts, Factorial, Binomial Coefficient. Provisions and combinations, count the injecting functions and functions. (6h)
Euclidean algorithm of subsequent divisions for the calculation of the MCD, prime numbers, fundamental theorem of the arithmetic (with demonstration), theorem on the existence of infinite prime numbers (with demonstration). (4h)
Numbering based on n. Congruence Report Module n. Solve linear congruences. Module n. (6h)
Operations on a set, commutative and associative properties, neutral element and invertible elements. Invertible elements in Zm, function of Euler. (4h)
Monoids and groups. Numerical and non-numeric examples (word monoid, permutation group). Definition of subgroup, examples. Example of the group of square arrays of order 2, with a non-zero determinant. Subgroups, equivalence relation determined by a subgroup, right side, Lagrange theorem. (4h)
Rings, examples (ring of the whole module n, ring of the matrices on R, ring of the polynomials). Divisors of 0 and invertible. Fields, examples (field of the real, field of complexes). (4h)
Matrices on a field, matrix operations. Determinant and Rank (Laplace and Sarrus method, Kronecker's method for rank). Inverse of an array. Reduction in triangular shape. (4h)
Linear equation systems (homogeneous and non-homogeneous): Gauss-Jordan method. Rouchè-Capelli Theorem, Cramer Theorem. (8h)
Definition of vector space and examples. Vector subsoil. Linearly independent set. Generated subsoil. Space of solutions of a homogeneous system. Basics and size of a vector space. Size of a subspace. (4h)
Linear applications: matrix associated with a linear application. Core and image of a linear application. Theorem of nullity more rank. (6h)
Eigenvalues and eigenvectors, geometric and algebraic multiplicity of an eigenvalue, bases formed by eigenvectors. (4h)
Course notes will be available on the web site.
Other suggested reading are:
• Introduzione alla Matematica Discreta, di M Bianchi e A. Gillio, McGraw-Hill.
• Elementi di Matematica Discreta e Algebra Lineare di F. Dalla Volta e M. Rigoli, Pearson Education, 2007.
Lectures and exercises