# BASIC MATHEMATICS

## Learning outcomes of the course unit

Formative purposes:

The aim of the course is to provide basic mathematical knowledge, mostly tackled in the programmes of primary and secondary schools, necessary to follow all of the courses of the first year. During the formative activity the student can fill possible gaps or consolidate his knowledge.

At the end of the course the student is expected to be able:

Knowledge and understanding:

- to know the different sets of numbers and their properties

- to have understood logic of propositions

- to remember all the properties of equations, inequalities and systems

- to know general function theory

- to know the basics of trigonometry

- to have understood the concepts of absolute value, exponential and logarithm

- to know basic figures of analytical geometry and their equations

- have understood the basic concepts of set theory and their relationship with the logic of propositions

- know the concept of cardinality and his relation with the properties of functions

- know the various types of induction and understand which is the most suitable for the problem they consider

- extend the concepts of analytic geometry to the understanding of the subsets of the Cartesian plane

Applying knowledge and understanding:

- to order numbers, to factorize a polynomial

- to perform calculations with fractions, radicals, exponentials and logarithms

- to calculate sine, cosine and tangent of a known angle

- to analyse and to deny a proposition

- to solve equations, inequalities and systems of first and second degree, of degree larger than two, with absolute value, irrational, trigonometric, exponential and logarithmic

- to determine the domain, the range and the inverse image of a function whose graph is given; to prove that a function is injective, surjective, increasing or decreasing

- to draw the graph of an elementary function or of a piecewise defined function based on transformations of elementary functions

- to analyse and draw straight lines, parabolas, circumferences, ellipses, hyperbolas

- prove the properties of sets, also starting from the intensional definition

- understand how the various properties of functions behave with respect to composition

- apply the properties of functions to the computation of cardinalities

- demonstrate propositions using the various forms of induction

- solve basic equations in two variables involving injective and surjective functions

- identify subsets of the Cartesian plane defined by equations, inequalities, systems in one and two variables

Making judgments:

- to be able to face an autonomous analysis of possible applications during the following courses

- understand the logical structure of simple statements and imagine a demonstrative path

Communication skills:

- to have acquired the ability to work both autonomously and in groups

Learning skills:

- to be able to carry on with his scientific studies autonomously.

## Prerequisites

Prerequisites:

Mandatory prerequisites: elementary mathematics.

On the Elly platform 20-21 in the “CONOSCENZE PRELIMINARI” section are available lectures and exercises on preliminary knowledge.

In addition, in the period 6th-10th September will take place a pre-course on the mathematical knowledge required for the course.

All further mathematical knowledge from primary and secondary schools is useful.

## Course contents summary

Topics:

PART 1 (3CFU – MATHEMATICS and PHYSICS):

LOGIC OF PROPOSITIONS

FUNCTIONS (GENERALITIES)

EXPONENTIAL FUNCTIONS, LOGARITHMS

ANALYTICAL GEOMETRY

PART 2 (3CFU – MATHEMATICS):

FUNCTIONS (DETAILED STUDY)

BASIC ELEMENTS OF NAIVE SET THEORY

CARDINALITY

MATHEMATICAL INDUCTION

EQUATIONS AND INEQUALITIES IN TWO VARIABLES, SUBSETS OF THE CARTESIAN PLANE

## Course contents

Topics:

PART 1 (3CFU – MATHEMATICS and PHYSICS):

EQUATIONS AND INEQUALITIES

Properties of equalities and inequalities, passage to the reciprocal, squaring.

Equations and inequalities of any degree, fractions and products, biquadratic equations and biquadratic inequalities. Systems.

PROPOSITIONAL LOGIC

Logic of propositions: definition of proposition, truth value, examples, logical connectives (not, and, or), truth tables, predicates, quantifiers (for each, exists, exists unique), negation of a proposition. Implication, examples, negation of an implication, counterexamples, necessary and sufficient condition, contrapositive implication. Biconditional connective and its negation, proofs by contradiction. Extended and abbreviated scriptures, negation of such scriptures.

Analysis of a proposition and its negation.

FUNCTIONS

Definition of function, domain, graph, image, image and counter-images of an element, injective, surjective, bijective functions.

Increasing and decreasing, monotone functions. Relation between monotonicity and injectivity.

Negation of all definitions.

Piecewise defined functions.

Operations on functions: addition, subtraction, multiplication, ratio, composition. Effects of these operations on injective, surjective, bijective and monotone functions.

Inverse function, graph of the inverse function.

Square root, cubic root functions.

Simple equations and irrational inequalities.

Even and odd functions.

Absolute value function. Simple equations and inequalities with absolute value.

Additional properties of functions.

ANALYTIC GEOMETRY

Review: lines and parables.

Circles: locus of points, center, radius, canonical equation.

Ellipses: locus of points, foci, center, vertices, semi-axes, canonical equation.

Hyperbola: locus of points, foci, center, vertices, asymptotes, canonical equation.

Intersections between the various conics.

EXPONENTIAL FUNCTIONS and LOGARITHMS

Exponential functions: trend and graph, solving simple equations and inequalities.

Logarithms: definition, simple operations with logarithms, change of base, graph of the logarithm with natural basis, solving simple equations and inequalities.

FUNCTIONS

Graph summary of all elementary functions, including the sine, cosine and tangent graph. Reconstruction of the equation of a parabola known the vertex.

Graphs from other graphs: translations, homothetics and symmetries, graph of the absolute value of a function |f (x)| and of f(|x|).

PART 2 (3CFU – MATHEMATICS):

FUNCTIONS: composition of injective, surjective, bijective maps.

ELEMENTS OF NAÏVE SET THEORY: definitions of union, intersection, empty set, difference, symmetric difference, Cartesian product, fundamental properties of these operators. Inclusion, subsets, power set, partitions. Set theory from a propositional point of view.

CARDINALITY: definition of cardinals, finite sets, equipotency, countability of rationals, Cantor's theorem (cardinality of the set of parts), uncountability of real numbers.

MATHEMATICAL INDUCTION: basic induction, variants. Summations.

EQUATIONS AND INEQUALITIES IN TWO VARIABLES: equations in two variables with injective and surjective functions. Subsets of the Cartesian plane defined by equations and inequalities in two variables, systems of equations and inequalities. Sub-graphs and epigraphs of functions. Review of conics.

## Recommended readings

Reference books:

E. Acerbi, G. Buttazzo: Matematica Preuniversitaria di Base. Pitagora Editrice, Bologna (2003).

F.G. Alessio, C. de Fabritiis, C. Marcelli, P. Montecchiari: Matematica zero, Pearson Italia, Milano-Torino (2016).

Additional material (on the ELLY platform, https://elly2021.smfi.unipr.it/, MATEMATICA, primo anno, ELEMENTI DI MATEMATICA):

Lectures and exercises on preliminary knowledge.

Lectures 2017-18.

Exercises with solution.

Examinations with solution (from 17-18 to 20-21).

## Teaching methods

Teaching methods:

The teaching activities will be performed in the University classrooms.

PART 1 (3CFU – MATHEMATICS and PHYSICS): 28 hours of lectures and practices.

The course takes place from the 13th of September to the 1st of October 2021, with 14 hours the first week, 8 hours in the second one and 6 hours in the third one.

The first two Fridays (17th and 27th September), 2 hours will be devoted to a formative test and guided exercises during which the students, autonomously or in small groups, solve the proposed exercises with the supervision of the teacher.

The two formative tests (each 45 minutes long) are planned in order to point out gaps, evaluate the progress of the student’s learning and give a feedback to the students before the final test. The results of those tests will not affect the final test.

The last Friday (1st of October) 4 hours will be devoted to the preparation of the exam.

The first exam on the FIRST PART is fixed on the 4th of October.

PART 2 (3CFU – MATHEMATICS): 28 hours of lectures and practices.

The course takes place from the 20th of September to the 12th of November 2021, with 2 hours the first two weeks and 4 hours the last six weeks. At the end of the course an exam on the SECOND PART is fixed.

Teachers will be available in a specific time period or by appointment to discuss lectured topics.

At the beginning of the course all the material of the FIRST PART is uploaded on the Elly platform, while the material of the SECOND PART is uploaded weekly.

In order to download the teaching stuff, the on-line registration is needed.

The uploaded lectures are an integral part of the teaching material.

Non-attending students are advised to check the available teaching material and the information given by the teacher via the Elly platform.

## Assessment methods and criteria

Method of testing learning:

The final evaluation on the learning consists of a 3 hours long written test (2 hours for Physics); books, notes and electronic devices are not allowed.

The exam will take place in presence in the lecture-hall.

The student must prove he/she has understood, and is able to apply, the basic concepts of every topic in the programme.

The written test is split into two parts: an applying knowledge test for all students in Mathematics and Physics (FIRST PART of the course) and a theoretical test only for students of Mathematics (SECOND PART of the course).

Instead of the final test, the student will be allowed to sustain two written tests in itinere, one for each part.

The final mark is calculated on 32 points. The exam is passed with a final mark of minimum 18/30.

The results of the exam will be published on the Elly platform within two weeks from the written test.

The students can examine their written tests during the time specified by the teacher or by appointment.

## Other informations

Other information:

This course (6CFU) is mandatory for all students in Mathematics.

The FIRST PART of this course (3CFU) is mandatory for all students in Physics.

The course of Basic Mathematics is a prerequisite to Algebra and Mathematical Analysis 1. Attending to the course is mandatory (75%) for all students who have not passed their self-evaluation test or are relieved from it, for instance with the certificate of the final test of the CORDA Project with at least 50 points. For the full list of relieving conditions see the OFA (added formative obligations) in the Manifesto degli studi 2021-22 at the entry: ESONERO TEST VPI. Moreover for these students the course of Basic Mathematics is a prerequisite to all other exams in the degree course of Mathematics.

Attending to the course is strongly recommended to all students