Elements of Basic Mathematics
A.Y. 2020/2021
Learning objectives
The aim of this course is to provide students with the basic language and the essential tools, which are the fundamentals to face the BSc program in Mathematics.
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
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.
(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.
Prerequisites for admission
No specific preliminary knowledge is required
Teaching methods
Frontal lessons. Weekly virtual meetings on either Zoom or Microsoft Teams, for students who need etra explanations.
Teaching Resources
1) Matematica zero- ROMA Casa Editrice Aracne (1984)
2) M.Bramanti-G.Travaglini: Matematica. Questione di Metodo -Zanichelli (2009)
2) M.Bramanti-G.Travaglini: Matematica. Questione di Metodo -Zanichelli (2009)
Assessment methods and Criteria
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 (handled by Syllabus Course teachers ): checking of the knowledge of Syllabus course subjects
Part B ( handled by EMB teachers): checking of the knowledge of EMB course subjects.
The complete final examination is passed if both part A and part B of the written exam are successfully passed. Final marks are given Approved-Not approved, and will be communicated on SIFA
Part A (handled by Syllabus Course teachers ): checking of the knowledge of Syllabus course subjects
Part B ( handled by EMB teachers): checking of the knowledge of EMB course subjects.
The complete final examination is passed if both part A and part B of the written exam are successfully passed. Final marks are given Approved-Not approved, and will be communicated on SIFA
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:
Mantovani Sandra
Elementi di matematica di base (ediz.2)
Responsible
Lesson period
First semester
Teaching methods:
The course will be held mainly by online asynchronous lectures uploaded on the Ariel web site of the course, according to the timeschedule of both the editions. Moreover tutoring meetings will be planned using the platform Zoom.
The course will be held mainly by online asynchronous lectures uploaded on the Ariel web site of the course, according to the timeschedule of both the editions. Moreover tutoring meetings will be planned using the platform Zoom.
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.
(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.
Prerequisites for admission
No specific preliminary knowledge is required
Teaching methods
Frontal lessons.
Teaching Resources
1) Matematica zero- ROMA Casa Editrice Aracne (1984)
2) M.Bramanti-G.Travaglini: Matematica. Questione di Metodo -Zanichelli (2009)
2) M.Bramanti-G.Travaglini: Matematica. Questione di Metodo -Zanichelli (2009)
Assessment methods and Criteria
It consists of two parts and successfulness in part A is a necessary condition for the correction of part B:
Part A ( handled by Syllabus Course teachers ): check of knowledge of Syllabus
course subjects
Part B (handled by EMB teachers): checking of the knowledge of EMB course subjects.
The complete final examination is passed if both part A and part B of the written exam are successfully passed. Final marks are given Approved-Not approved, and will be communicated on SIFA
Part A ( handled by Syllabus Course teachers ): check of knowledge of Syllabus
course subjects
Part B (handled by EMB teachers): checking of the knowledge of EMB course subjects.
The complete final examination is passed if both part A and part B of the written exam are successfully passed. Final marks are given Approved-Not approved, and will be communicated on SIFA
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:
Bertolini Marina
Professor(s)
Reception:
Thursday 12.45-14.15, by appointment
Studio 1019, I Floor, Dipartimento di Matematica, Via Saldini, 50