Mathematics I
A.Y. 2026/2027
Learning objectives
The primary goals of this course are to introduce the language and fundamental concepts of discrete mathematics, algebra, and real analysis of a single variable. In more detail, the course will cover the elementary theory of sets, relations, functions, and real-valued functions of a single real variable. A first introduction to the notion of abstract algebraic structures is provided using monoids, groups, and rings as examples. The basic properties of the ring of integers and the fields of rational, real, and complex numbers are discussed, focusing on the resolution of linear congruences and their algorithmic aspects. Elementary operations with complex numbers are introduced for solving first- and second-degree equations. Vector spaces are also treated, along with linear transformations and their matrix representation. The theory of linear algebra is then applied to the solution of systems of linear equations, highlighting their algorithmic aspects as well.
Expected learning outcomes
At the end of the course, students should be able to understand the basic mathematical formalism of sets, relations, and functions. They will know how to solve simple exercises concerning real-valued functions of a single real variable, such as calculating the domain and representing the graph of elementary functions. They will have acquired a foundational familiarity with the concept of abstract algebraic structures. They will have understood the fundamental properties of the ring of integers and the field of rational, real, and complex numbers. They will have acquired familiarity with the operations on complex numbers and be able to calculate powers and roots of complex numbers. They will be able to recognize and work with vector spaces and linear transformations. Finally, they will be capable of performing operations with matrices, associating them with linear systems, and using them to analyze their solvability.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
Part 1. Preliminary concepts.
Combinatorics: how to count (outline). Sets and cardinality. The algebra of sets: subsets, set of parts, operations on subsets, sum and product of sets. Natural, integer, rational, and real numbers. Equations and inequalities, intervals of R. The axiom of completeness of the real numbers. The set of rational numbers is not complete. The existence of the square root of 2 in the real numbers. Maxima and minima of subsets of R. Majorities and minorities. Upper and lower bounds of subsets of R.
Part 2. Relations, functions, and structures.
2.1 Relations and functions.
Relations and functions between sets. Equivalence relations and partitions. Partial and linear order relations. Composition of relations and functions. Inverse relation. Injective, surjective, and bijective functions. Identity function, inverse of a function, isomorphisms. Image and counterimage of a set with respect to a function. Kernel of a function.
2.2 Real functions with one real variable.
Graph, natural domain, and restriction of a function. Monotone functions. Sum, product, and ratio of functions. Symmetries. Linear functions and absolute value. Power functions. Exponential functions and logarithms. The trigonometric functions sine, cosine, and tangent. Calculating some notable values of sin(x), cos(x), and tan(x). Addition and duplication formulas. Inverse trigonometric functions: arccosine, arcsine, arctangent.
2.3 Permutations and Groups.
Symmetric group. Factorization of permutations into cycles and transpositions. Period and sign of permutations. Alternating group. Groups (notes).
2.4 Divisibility and Rings.
Integers and divisibility. Prime numbers and factorization. Euclidean algorithm. Bézout's identity. Linear Diophantine equations in two variables. Modular arithmetic and linear congruences (an overview). Polynomials and divisibility. Division between polynomials. Irreducible polynomials and factorization. Rings (an overview).
2.5 Complex Numbers and Fields.
Operations with complex numbers. Cartesian representation. Inverse, conjugate, and modulus of a complex number. Trigonometric form of complex numbers. Product formula. De Moivre's formula for powers of complex numbers. Nth roots of complex numbers. Fundamental theorem of algebra. Quadratic equations. Fields (an overview).
Part 3. Linear algebra.
Systems of m linear equations in n unknowns. Resolution with the Gauss-Jordan echelon reduction method. Matrices and their algebra. Vector spaces and subspaces (an overview). Foundations. Determinant of a square matrix and its properties. Invertible matrices. Computing the inverse matrix. Matrix rank; matrices and linear applications. Cramer and Rouché-Capelli theorems. Eigenvalues and eigenspaces.
Each topic in the program is accompanied by examples, exercises, and applications.
Combinatorics: how to count (outline). Sets and cardinality. The algebra of sets: subsets, set of parts, operations on subsets, sum and product of sets. Natural, integer, rational, and real numbers. Equations and inequalities, intervals of R. The axiom of completeness of the real numbers. The set of rational numbers is not complete. The existence of the square root of 2 in the real numbers. Maxima and minima of subsets of R. Majorities and minorities. Upper and lower bounds of subsets of R.
Part 2. Relations, functions, and structures.
2.1 Relations and functions.
Relations and functions between sets. Equivalence relations and partitions. Partial and linear order relations. Composition of relations and functions. Inverse relation. Injective, surjective, and bijective functions. Identity function, inverse of a function, isomorphisms. Image and counterimage of a set with respect to a function. Kernel of a function.
2.2 Real functions with one real variable.
Graph, natural domain, and restriction of a function. Monotone functions. Sum, product, and ratio of functions. Symmetries. Linear functions and absolute value. Power functions. Exponential functions and logarithms. The trigonometric functions sine, cosine, and tangent. Calculating some notable values of sin(x), cos(x), and tan(x). Addition and duplication formulas. Inverse trigonometric functions: arccosine, arcsine, arctangent.
2.3 Permutations and Groups.
Symmetric group. Factorization of permutations into cycles and transpositions. Period and sign of permutations. Alternating group. Groups (notes).
2.4 Divisibility and Rings.
Integers and divisibility. Prime numbers and factorization. Euclidean algorithm. Bézout's identity. Linear Diophantine equations in two variables. Modular arithmetic and linear congruences (an overview). Polynomials and divisibility. Division between polynomials. Irreducible polynomials and factorization. Rings (an overview).
2.5 Complex Numbers and Fields.
Operations with complex numbers. Cartesian representation. Inverse, conjugate, and modulus of a complex number. Trigonometric form of complex numbers. Product formula. De Moivre's formula for powers of complex numbers. Nth roots of complex numbers. Fundamental theorem of algebra. Quadratic equations. Fields (an overview).
Part 3. Linear algebra.
Systems of m linear equations in n unknowns. Resolution with the Gauss-Jordan echelon reduction method. Matrices and their algebra. Vector spaces and subspaces (an overview). Foundations. Determinant of a square matrix and its properties. Invertible matrices. Computing the inverse matrix. Matrix rank; matrices and linear applications. Cramer and Rouché-Capelli theorems. Eigenvalues and eigenspaces.
Each topic in the program is accompanied by examples, exercises, and applications.
Prerequisites for admission
Familiarity with basic high-school mathematics.
Teaching methods
Lectures and exercises.
Teaching Resources
Some useful reference textbooks for the various parts of the course are:
- P. Marcellini e C. Sbordone, Elementi di Analisi Matematica uno, Liguori, 2002.
- G. Piacentini Cattaneo, Matematica discreta e applicazioni, Zanichelli, 2008.
- R. Fioresi e M. Morigi, Introduction to Linear Algebra, CEA, 2019.
Further material is available a the course's website.
- P. Marcellini e C. Sbordone, Elementi di Analisi Matematica uno, Liguori, 2002.
- G. Piacentini Cattaneo, Matematica discreta e applicazioni, Zanichelli, 2008.
- R. Fioresi e M. Morigi, Introduction to Linear Algebra, CEA, 2019.
Further material is available a the course's website.
Assessment methods and Criteria
The course includes a written exam, usually two and a half hours long, typically consisting of six exercises on course topics. There is no oral exam. Two written midterm tests are planned for.
MATH-02/B - Geometry - University credits: 4
MATH-03/A - Mathematical Analysis - University credits: 5
MATH-03/A - Mathematical Analysis - University credits: 5
Exercises: 36 hours
Lessons: 48 hours
Lessons: 48 hours
Professors:
Marra Vincenzo, Pertusi Laura
Professor(s)
Reception:
By appointment
Dipartimento di Matematica "Federigo Enriques", via Cesare Saldini 50, room 2048