Elements of Basic Mathematics
A.Y. 2018/2019
Learning objectives
The aim of this course is to provide students with the basic language and the essential tools of mathematics which are the fundamentals to face the first level graduation program.
Expected learning outcomes
After this course, the students should be able to manage independently elementary concepts of logic, of elementary set theory and functions, and of real numbers.
Lesson period: First semester
Assessment methods: Giudizio di approvazione
Assessment result: superato/non superato
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Elementi di matematica di base (ediz. 1)
Responsible
Lesson period
First semester
Course syllabus
(1) Basics of Logic: Compound propositions and logical connectives: conjuction, disjunction, negation, implication, biconditional. Truth tables. Logical implication and equivalence, necessary and/or sufficient conditions, universal quantifier, existential quantifier and their negation. Proofs by contrapositive and by contradiction.
(2) Basics of set theory: Relation of membership. Elements and subsets of a set. Inclusion, union, intersection, complement; power set and Cartesian product; relations and functions (injective, surjective, bijective), graphs of functions (several examples, among them elementary functions and piecewise functions). Partition of a set. Equivalence relations and quotient sets. Natural numbers and mathematical induction principle. Cardinality of the power set of a finite set.
(3) Rational numbers and their representation; real numbers. Rational numbers as fractions, this representation is not unique, fraction in its lowest terms. Representation of the rational numbers as points on an oriented line. Representation of the rational numbers as finite decimal or infinite periodic sequences. Non-existence of a rational number with square 2. Real numbers as infinite decimal sequences. Bounded and unbounded subsets in R. Sup/Inf, maximum and minimum of a set in R. Intervals in R and their notation.
EXAM
It consists of two parts (to be done in the same day), successfulness in part A is a necessary condition for the correction of part B:
Part A (1 hour, handled by Syllabus Course teachers ): check of knowledge of Syllabus
course subjects
Part B (2 hours, handled by EMB teachers): check of knowledge of EMB course subjects.
The exam is only for first year students.
(2) Basics of set theory: Relation of membership. Elements and subsets of a set. Inclusion, union, intersection, complement; power set and Cartesian product; relations and functions (injective, surjective, bijective), graphs of functions (several examples, among them elementary functions and piecewise functions). Partition of a set. Equivalence relations and quotient sets. Natural numbers and mathematical induction principle. Cardinality of the power set of a finite set.
(3) Rational numbers and their representation; real numbers. Rational numbers as fractions, this representation is not unique, fraction in its lowest terms. Representation of the rational numbers as points on an oriented line. Representation of the rational numbers as finite decimal or infinite periodic sequences. Non-existence of a rational number with square 2. Real numbers as infinite decimal sequences. Bounded and unbounded subsets in R. Sup/Inf, maximum and minimum of a set in R. Intervals in R and their notation.
EXAM
It consists of two parts (to be done in the same day), successfulness in part A is a necessary condition for the correction of part B:
Part A (1 hour, handled by Syllabus Course teachers ): check of knowledge of Syllabus
course subjects
Part B (2 hours, handled by EMB teachers): check of knowledge of EMB course subjects.
The exam is only for first year students.
MAT/01 - MATHEMATICAL LOGIC
MAT/02 - ALGEBRA
MAT/03 - GEOMETRY
MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS
MAT/05 - MATHEMATICAL ANALYSIS
MAT/06 - PROBABILITY AND STATISTICS
MAT/07 - MATHEMATICAL PHYSICS
MAT/08 - NUMERICAL ANALYSIS
MAT/09 - OPERATIONS RESEARCH
MAT/02 - ALGEBRA
MAT/03 - GEOMETRY
MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS
MAT/05 - MATHEMATICAL ANALYSIS
MAT/06 - PROBABILITY AND STATISTICS
MAT/07 - MATHEMATICAL PHYSICS
MAT/08 - NUMERICAL ANALYSIS
MAT/09 - OPERATIONS RESEARCH
Lessons: 27 hours
Professor:
Marra Vincenzo
Elementi di matematica di base (ediz. 2)
Lesson period
First semester
Course syllabus
(1) Basics of Logic: Compound propositions and logical connectives: conjuction, disjunction, negation, implication, biconditional. Truth tables. Logical implication and equivalence, necessary and/or sufficient conditions, universal quantifier, existential quantifier and their negation. Proofs by contrapositive and by contradiction.
(2) Basics of set theory: Relation of membership. Elements and subsets of a set. Inclusion, union, intersection, complement; power set and Cartesian product; relations and functions (injective, surjective, bijective), graphs of functions (several examples, among them elementary functions and piecewise functions). Partition of a set. Equivalence relations and quotient sets. Natural numbers and mathematical induction principle. Cardinality of the power set of a finite set.
(3) Rational numbers and their representation; real numbers. Rational numbers as fractions, this representation is not unique, fraction in its lowest terms. Representation of the rational numbers as points on an oriented line. Representation of the rational numbers as finite decimal or infinite periodic sequences. Non-existence of a rational number with square 2. Real numbers as infinite decimal sequences. Bounded and unbounded subsets in R. Sup/Inf, maximum and minimum of a set in R. Intervals in R and their notation.
EXAM
It consists of two parts (to be done in the same day), successfulness in part A is a necessary condition for the correction of part B:
Part A (1 hour, handled by Syllabus Course teachers ): check of knowledge of Syllabus
course subjects
Part B (2 hours, handled by EMB teachers): check of knowledge of EMB course subjects.
The exam is only for first year students.
(2) Basics of set theory: Relation of membership. Elements and subsets of a set. Inclusion, union, intersection, complement; power set and Cartesian product; relations and functions (injective, surjective, bijective), graphs of functions (several examples, among them elementary functions and piecewise functions). Partition of a set. Equivalence relations and quotient sets. Natural numbers and mathematical induction principle. Cardinality of the power set of a finite set.
(3) Rational numbers and their representation; real numbers. Rational numbers as fractions, this representation is not unique, fraction in its lowest terms. Representation of the rational numbers as points on an oriented line. Representation of the rational numbers as finite decimal or infinite periodic sequences. Non-existence of a rational number with square 2. Real numbers as infinite decimal sequences. Bounded and unbounded subsets in R. Sup/Inf, maximum and minimum of a set in R. Intervals in R and their notation.
EXAM
It consists of two parts (to be done in the same day), successfulness in part A is a necessary condition for the correction of part B:
Part A (1 hour, handled by Syllabus Course teachers ): check of knowledge of Syllabus
course subjects
Part B (2 hours, handled by EMB teachers): check of knowledge of EMB course subjects.
The exam is only for first year students.
MAT/01 - MATHEMATICAL LOGIC
MAT/02 - ALGEBRA
MAT/03 - GEOMETRY
MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS
MAT/05 - MATHEMATICAL ANALYSIS
MAT/06 - PROBABILITY AND STATISTICS
MAT/07 - MATHEMATICAL PHYSICS
MAT/08 - NUMERICAL ANALYSIS
MAT/09 - OPERATIONS RESEARCH
MAT/02 - ALGEBRA
MAT/03 - GEOMETRY
MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS
MAT/05 - MATHEMATICAL ANALYSIS
MAT/06 - PROBABILITY AND STATISTICS
MAT/07 - MATHEMATICAL PHYSICS
MAT/08 - NUMERICAL ANALYSIS
MAT/09 - OPERATIONS RESEARCH
Lessons: 27 hours
Professor:
Bianchi Mariagrazia
Professor(s)
Reception:
By appointment
Dipartimento di Matematica "Federigo Enriques", via Cesare Saldini 50, room 2048