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Algebra e Logica
Algebra and Logic
Maria Chiara Brambilla

Seat Ingegneria
A.A. 2016/2017
Credits 6
Hours 48
Period II
Language ENG

Prerequisites
Basic notions of Linear Algebra

Learning outcomes
KNOWLEDGE AND UNDERSTANDING:
The course will teach to students basic notions of algebra and logic. In particular the course will concern the notions of sets, equivalence relations, order relations, groups, rings, fields; logic thought and basic rules of proposition and predicative logic.
CAPACITY TO APPLY KNOWLEDGE AND UNDERSTANDING:
Students will be able to use the course topics, to provide correct proofs, to solve exercises on groups, rings and fields, to read and write well posed expressions in logic and to compute the true value of a logic expression.
TRANSVERSAL SKILLS:
The study and the exercises proposed in this course will consolidate also the “transverse skills” obtained by the students. In particular the ability of critically analyze a problem and the ability of use a correct, precise and formal language. The ability to learn in autonomy and to develop the study will be strenghtened too.

Program
Set theory. Arithmetic: division, prime numbers, GCD, Bezout's identity, Euclid's algorithm. Equivalence relations, quotients, integers modulo n. Order relations and lattices. Groups, subgroups, group homomorphisms, quotient group. Lagrange Theorem. Rings and fields, integral domains, ideals, quotient ring. Polynomial rings. Introduction to mathematical logic. Propositional logic: syntax, semantics, natural deduction system. Soundness and completeness theorems. First-order logic: syntax and semantics.

Development of the examination
LEARNING EVALUATION METHODS
The learning evaluation is carried out by two exams: - a practical examination, which consists of solving exercises and problems related to the topics explained in the course. - a theoretical examination, consisting in a discussion of the topics of the course. In particular the knowledge and the understanding of all definitions, theorems and proofs explained in the classes will be tested. The practical exam is preliminary to the theoretical one. It is necessary to pass the practical exam in order to do the theoretical one.

LEARNING EVALUATION CRITERIA
In order to pass the learning evaluation, the student must demonstrate that he/she has understood the basic concepts of algebra and logic explained in the course. In particular in the practical test the student must show that he/she is able to apply independently the learned techniques in solving exercises and problems. In the theoretical exam the student must be able to expose the theoretical contents with the correct language and accuracy.

LEARNING MEASUREMENT CRITERIA
Each of the tests is graded on a scale from 0 to 30. The final grade will be decided starting from the two test grades.

FINAL MARK ALLOCATION CRITERIA
The final grade will be positive only if in both of the tests the students gets the passing grade (18/30). The maximal grade is reached if the student proves a knowledge and a thorough understanding of the course content. The maximal grade with honors is reserved to the students who passed both of the tests in a complet and correct way, showing special independence and excellence.

Recommended reading
Facchini, “Algebra e matematica discreta“, Zanichelli. Asperti, Ciabattoni, “Logica a informatica“, McGraw-Hill.

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