Discrete Mathematics
A.Y. 2024/2025
Learning objectives
The course has the main purposes of intoducing the algebraic language and the notions of vector spaces and linear applications and of analyzing the problem of resolving linear systems of equations (even from an algorithm point of view).
Expected learning outcomes
The student should be able to understant the formal language of the abstract algebra, to discuss the resolution of linear systems, to recognise the vector spaces and the linear applications. Moreover, he should be able to work with the matrices, to associate them to linear systems and to discuss their diagonalization.
Lesson period: Second 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
Edition 1
Responsible
Lesson period
Second semester
Course syllabus
1) Basic algebraic tools and algorithms
Integers: induction; division; Euclidean algorithm; prime numbers; integer factorization.
Polynomials with real coefficients. Roots. Irreducible polynomials. Factorization of polynomials.
Linear systems; Gauss-Jordan method.
Matrices and their algebra.
2) An outline of abstract algebra
Sets and relations: equivalence relations, partial orderings,maps. Congruence: the integers mod n.
Algebraic structures and their homomorphisms: groups, rings (fields and polynomial rings).
3) Linear Algebra
Vector spaces. Bases. Linear maps and matrices; the rank of a matrix. Determinants of square matrices. Inverse matrices: existence and computing. Cramer and Rouché-Capelli theorems. Eigenvalues and eigenvectors.
The course will be supported by practical exercises to improve comprehension of the several subjects discussed during lectures.
[Program for not attending students with reference to descriptor 1 and 2]:
1) Basic algebraic tools and algorithms
Integers: induction; division; Euclidean algorithm; prime numbers; integer factorization.
Polynomials with real coefficients. Roots. Irreducible polynomials. Factorization of polynomials.
Linear systems; Gauss-Jordan method.
Matrices and their algebra.
2) An outline of abstract algebra
Sets and relations: equivalence relations, partial orderings,maps. Congruence: the integers mod n.
Algebraic structures and their homomorphisms: groups, rings (fields and polynomial rings).
3) Linear Algebra
Vector spaces. Bases. Linear maps and matrices; the rank of a matrix. Determinants of square matrices. Inverse matrices: existence and computing. Cramer and Rouché-Capelli theorems. Eigenvalues and eigenvectors.
Integers: induction; division; Euclidean algorithm; prime numbers; integer factorization.
Polynomials with real coefficients. Roots. Irreducible polynomials. Factorization of polynomials.
Linear systems; Gauss-Jordan method.
Matrices and their algebra.
2) An outline of abstract algebra
Sets and relations: equivalence relations, partial orderings,maps. Congruence: the integers mod n.
Algebraic structures and their homomorphisms: groups, rings (fields and polynomial rings).
3) Linear Algebra
Vector spaces. Bases. Linear maps and matrices; the rank of a matrix. Determinants of square matrices. Inverse matrices: existence and computing. Cramer and Rouché-Capelli theorems. Eigenvalues and eigenvectors.
The course will be supported by practical exercises to improve comprehension of the several subjects discussed during lectures.
[Program for not attending students with reference to descriptor 1 and 2]:
1) Basic algebraic tools and algorithms
Integers: induction; division; Euclidean algorithm; prime numbers; integer factorization.
Polynomials with real coefficients. Roots. Irreducible polynomials. Factorization of polynomials.
Linear systems; Gauss-Jordan method.
Matrices and their algebra.
2) An outline of abstract algebra
Sets and relations: equivalence relations, partial orderings,maps. Congruence: the integers mod n.
Algebraic structures and their homomorphisms: groups, rings (fields and polynomial rings).
3) Linear Algebra
Vector spaces. Bases. Linear maps and matrices; the rank of a matrix. Determinants of square matrices. Inverse matrices: existence and computing. Cramer and Rouché-Capelli theorems. Eigenvalues and eigenvectors.
Prerequisites for admission
Basic knowledge of mathematics, like solving equations and polynomial algebra.
Teaching methods
Frontal lectures about theory and classes of exercises.
Tutoring .
Attendance to theory and exercises classes is strongly recommended
Tutoring .
Attendance to theory and exercises classes is strongly recommended
Teaching Resources
Mainly: Ariel web page and notes of the course.
Suggested books:
M. Bianchi, A. Gillio - Introduzione alla matematica discreta - McGraw Hill (seconda edizione 2005).
A. Alzati, M. Bianchi, M. Cariboni - Matematica discreta - Esercizi - Pearson Education - (2006).
Suggested books:
M. Bianchi, A. Gillio - Introduzione alla matematica discreta - McGraw Hill (seconda edizione 2005).
A. Alzati, M. Bianchi, M. Cariboni - Matematica discreta - Esercizi - Pearson Education - (2006).
Assessment methods and Criteria
The final examination consists of a written test.
The written test will be comprised of exercises designed to test the ability to solve mathematical problems pertaining to the course syllabus, along with multiple-choice or true/false questions, and open-ended questions that will require the student to illustrate the proof of one of the theorems disccused in the course. The teachers will clearly indicate during the lectures which of the proofs presented in the course are examinable.
The duration of each written exam is commensurate with the number, structure, and difficulty of the exercises and questions assigned, but indicatively the exam is expected to last two and a half hours. During the semester in which the course is taught, there will also be a two-hour midterm which, if passed, will entitle the student to take fewer exercises in the written tests of the first two available appeals. The outcomes of the written and midterm exams will be communicated on SIFA through the UNIMIA portal. The final exam grade will be expressed on a scale from 0 to 30 with integer increments; 18 is the minimum passing grade.
The written test will be comprised of exercises designed to test the ability to solve mathematical problems pertaining to the course syllabus, along with multiple-choice or true/false questions, and open-ended questions that will require the student to illustrate the proof of one of the theorems disccused in the course. The teachers will clearly indicate during the lectures which of the proofs presented in the course are examinable.
The duration of each written exam is commensurate with the number, structure, and difficulty of the exercises and questions assigned, but indicatively the exam is expected to last two and a half hours. During the semester in which the course is taught, there will also be a two-hour midterm which, if passed, will entitle the student to take fewer exercises in the written tests of the first two available appeals. The outcomes of the written and midterm exams will be communicated on SIFA through the UNIMIA portal. The final exam grade will be expressed on a scale from 0 to 30 with integer increments; 18 is the minimum passing grade.
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
Practicals: 24 hours
Lessons: 32 hours
Lessons: 32 hours
Professors:
Oestvaer Paul Arne, Svaldi Roberto
Edition 2
Responsible
Lesson period
Second semester
Course syllabus
1) Basic algebraic tools and algorithms
Integers: induction; division; Euclidean algorithm; prime numbers; integer factorization.
Polynomials with real coefficients. Roots. Irreducible polynomials. Factorization of polynomials.
Linear systems; Gauss-Jordan method.
Matrices and their algebra.
2) An outline of abstract algebra
Sets and relations: equivalence relations, partial orderings,maps. Congruence: the integers mod n.
Algebraic structures and their homomorphisms: groups, rings (fields and polynomial rings).
3) Linear Algebra
Vector spaces. Bases. Linear maps and matrices; the rank of a matrix. Determinants of square matrices. Inverse matrices: existence and computing. Cramer and Rouché-Capelli theorems. Eigenvalues and eigenvectors.
The course will be supported by practical exercises to improve comprehension of the several subjects discussed during lectures.
[Program for not attending students with reference to descriptor 1 and 2]:
1) Basic algebraic tools and algorithms
Integers: induction; division; Euclidean algorithm; prime numbers; integer factorization.
Polynomials with real coefficients. Roots. Irreducible polynomials. Factorization of polynomials.
Linear systems; Gauss-Jordan method.
Matrices and their algebra.
2) An outline of abstract algebra
Sets and relations: equivalence relations, partial orderings,maps. Congruence: the integers mod n.
Algebraic structures and their homomorphisms: groups, rings (fields and polynomial rings).
3) Linear Algebra
Vector spaces. Bases. Linear maps and matrices; the rank of a matrix. Determinants of square matrices. Inverse matrices: existence and computing. Cramer and Rouché-Capelli theorems. Eigenvalues and eigenvectors.
Integers: induction; division; Euclidean algorithm; prime numbers; integer factorization.
Polynomials with real coefficients. Roots. Irreducible polynomials. Factorization of polynomials.
Linear systems; Gauss-Jordan method.
Matrices and their algebra.
2) An outline of abstract algebra
Sets and relations: equivalence relations, partial orderings,maps. Congruence: the integers mod n.
Algebraic structures and their homomorphisms: groups, rings (fields and polynomial rings).
3) Linear Algebra
Vector spaces. Bases. Linear maps and matrices; the rank of a matrix. Determinants of square matrices. Inverse matrices: existence and computing. Cramer and Rouché-Capelli theorems. Eigenvalues and eigenvectors.
The course will be supported by practical exercises to improve comprehension of the several subjects discussed during lectures.
[Program for not attending students with reference to descriptor 1 and 2]:
1) Basic algebraic tools and algorithms
Integers: induction; division; Euclidean algorithm; prime numbers; integer factorization.
Polynomials with real coefficients. Roots. Irreducible polynomials. Factorization of polynomials.
Linear systems; Gauss-Jordan method.
Matrices and their algebra.
2) An outline of abstract algebra
Sets and relations: equivalence relations, partial orderings,maps. Congruence: the integers mod n.
Algebraic structures and their homomorphisms: groups, rings (fields and polynomial rings).
3) Linear Algebra
Vector spaces. Bases. Linear maps and matrices; the rank of a matrix. Determinants of square matrices. Inverse matrices: existence and computing. Cramer and Rouché-Capelli theorems. Eigenvalues and eigenvectors.
Prerequisites for admission
Basic knowledge of mathematics, like solving equations and polynomial algebra.
Teaching methods
Frontal lectures about theory and classes of exercises.
Tutoring .
Attendance to theory and exercises classes is strongly recommended
Tutoring .
Attendance to theory and exercises classes is strongly recommended
Teaching Resources
Mainly: Ariel web page and notes of the course.
Suggested books:
M. Bianchi, A. Gillio - Introduzione alla matematica discreta - McGraw Hill (seconda edizione 2005).
A. Alzati, M. Bianchi, M. Cariboni - Matematica discreta - Esercizi - Pearson Education - (2006).
Suggested books:
M. Bianchi, A. Gillio - Introduzione alla matematica discreta - McGraw Hill (seconda edizione 2005).
A. Alzati, M. Bianchi, M. Cariboni - Matematica discreta - Esercizi - Pearson Education - (2006).
Assessment methods and Criteria
The final examination consists of a written test.
The written test will be comprised of exercises designed to test the ability to solve mathematical problems pertaining to the course syllabus, along with multiple-choice or true/false questions, and open-ended questions that will require the student to illustrate the proof of one of the theorems disccused in the course. The teachers will clearly indicate during the lectures which of the proofs presented in the course are examinable.
The duration of each written exam is commensurate with the number, structure, and difficulty of the exercises and questions assigned, but indicatively the exam is expected to last two and a half hours. During the semester in which the course is taught, there will also be a two-hour midterm which, if passed, will entitle the student to take fewer exercises in the written tests of the first two available appeals. The outcomes of the written and midterm exams will be communicated on SIFA through the UNIMIA portal. The final exam grade will be expressed on a scale from 0 to 30 with integer increments; 18 is the minimum passing grade.
The written test will be comprised of exercises designed to test the ability to solve mathematical problems pertaining to the course syllabus, along with multiple-choice or true/false questions, and open-ended questions that will require the student to illustrate the proof of one of the theorems disccused in the course. The teachers will clearly indicate during the lectures which of the proofs presented in the course are examinable.
The duration of each written exam is commensurate with the number, structure, and difficulty of the exercises and questions assigned, but indicatively the exam is expected to last two and a half hours. During the semester in which the course is taught, there will also be a two-hour midterm which, if passed, will entitle the student to take fewer exercises in the written tests of the first two available appeals. The outcomes of the written and midterm exams will be communicated on SIFA through the UNIMIA portal. The final exam grade will be expressed on a scale from 0 to 30 with integer increments; 18 is the minimum passing grade.
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
Practicals: 24 hours
Lessons: 32 hours
Lessons: 32 hours
Professors:
Camere Chiara, Carai Luca
Professor(s)
Reception:
By appointment (to be scheduled via email)
Department of Mathematics, via C. Saldini 50, second floor, office 2090
Reception:
By appointment (to be agreed upon via email)
Room 2102, Dipartimento di Matematica "F. Enriques", Via Saldini 50