#
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