#
Elements of Basic Mathematics

A.Y. 2017/2018

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
(In case of multiple editions, please check the period, as it may vary)

**Assessment methods:**

**Assessment result:**

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 - University credits: 0

MAT/02 - ALGEBRA - University credits: 0

MAT/03 - GEOMETRY - University credits: 0

MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 0

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 0

MAT/06 - PROBABILITY AND STATISTICS - University credits: 0

MAT/07 - MATHEMATICAL PHYSICS - University credits: 0

MAT/08 - NUMERICAL ANALYSIS - University credits: 0

MAT/09 - OPERATIONS RESEARCH - University credits: 0

MAT/02 - ALGEBRA - University credits: 0

MAT/03 - GEOMETRY - University credits: 0

MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 0

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 0

MAT/06 - PROBABILITY AND STATISTICS - University credits: 0

MAT/07 - MATHEMATICAL PHYSICS - University credits: 0

MAT/08 - NUMERICAL ANALYSIS - University credits: 0

MAT/09 - OPERATIONS RESEARCH - University credits: 0

Lessons: 27 hours

Professor:
Mantovani Sandra

### Elementi di matematica di base (ediz. 2)

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 - University credits: 0

MAT/02 - ALGEBRA - University credits: 0

MAT/03 - GEOMETRY - University credits: 0

MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 0

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 0

MAT/06 - PROBABILITY AND STATISTICS - University credits: 0

MAT/07 - MATHEMATICAL PHYSICS - University credits: 0

MAT/08 - NUMERICAL ANALYSIS - University credits: 0

MAT/09 - OPERATIONS RESEARCH - University credits: 0

MAT/02 - ALGEBRA - University credits: 0

MAT/03 - GEOMETRY - University credits: 0

MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 0

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 0

MAT/06 - PROBABILITY AND STATISTICS - University credits: 0

MAT/07 - MATHEMATICAL PHYSICS - University credits: 0

MAT/08 - NUMERICAL ANALYSIS - University credits: 0

MAT/09 - OPERATIONS RESEARCH - University credits: 0

Lessons: 27 hours

Professor:
Bianchi Mariagrazia

Professor(s)

Reception:

Thursday 12.45-14.15, by appointment

Studio 1019, I Floor, Dipartimento di Matematica, Via Saldini, 50