Programme

Elements of set theory. Maps between sets: invective, surjective, bijective maps.
Elements of propositional logic, truth tables. Equivalence and order relations.
Combinatorics. Binomial coefficients and binomial theorem. Permutations. The Integers: divisibility, GCD and Euclidean algorithm, Bézout identity, linear congruences.
Baiscs of algebraic structures: permutation groups, abstract groups, polynomials and finite fields.
Elements of graph theory.Lattices and Boolean algebras

Core Documentation

Giulia Maria Piacentini Cattaneo
Matematica discreta e applicazioni
Zanichelli 2008

Reference Bibliography

none

Type of delivery of the course

lectures

Attendance

recommended attendance

Type of evaluation

written test

Programme

Elements of Set Theory.
Union, intersection, Cartesian product, set subtraction, complementary set, cardinality.

Set functions.
Domain, codomain, Range. Injective, surjective, bijective functions. Inverse function, Identity, Permutations.

Elements of logic.
Propositional calculus, Operations between propositions.

Relations.
Reflexive, symmetric, antisymmetric, transitive. Order and equivalence relations. Examples.
Equivalence relations and classes. Quotient set.

Integer numbers, Division and its properties.
Greatest common divisor. Euclidean algorithm. Prime numbers. Fundamental theorem of arithmetic.

Congruence mod n.
Basic modular arithmetic. Sum and multiplication in Zn. Linear congruence. Description of linear congruence solutions. Euler's totient function. Small Fermat Theorem, Euler’s Theorem

Combinatory algebra.
Dispositions and combinations with and without repetitions, binomial coefficient. Properties. Tartaglia's triangle.

Partially ordered sets
Hasse Diagram. Maximum and minimum, Sup and inf. Reticular formations. Properties of inf e sup. Boolean Algebra. The Boolean operators AND, OR and NOT.

Core Documentation

Giulia Maria Piacentini Cattaneo
"Matematica discreta"
Edito da Zanichelli.

Type of delivery of the course

Classroom Teaching

Attendance

Not mandatory

Type of evaluation

A written test with questions regarding the theory and exercises, plus a possible oral test.

Programme

Lattices, Boolean algebras

Core Documentation

Giulia Maria Piacentini Cattaneo
Matematica discreta e applicazioni
Zanichelli 2008

Reference Bibliography

none

Type of delivery of the course

lectures

Attendance

recommended attendance

Type of evaluation

written test

Programme

1. Linear equations and numbers
Systems of linear equations. Represent linear systems with matrix equations. Equivalent systems. Natural, integers, rational, real numbers and their properties. Pills of Graph Theory: subsets and set subtraction.

2. Matrixes and sets
Matrix with real coefficients. Square, triangular, diagonal matrixes. Transposed and symmetric matrix. Pills of Graph Theory: union and intersection.

3. Vector spaces and matrices
Matrix addition and scalar product and their properties.

4. Matrix product
Matrix product and its mathematical properties: associative and distributive. Example of how matrix product does not satisfy the commutative and simplification properties. Matrices and linear systems

5. Determinants
Formal definition of determinant and its properties. Transposed and triangular matrix determinants. Binet’s theorem.

6. Inverse of a matrix
Identity matrix. Inverse of a matrix and its properties. Cramer’s theorem.

7. Matrix rank
Definition and properties. Minor of a matrix.

8. System of linear equations.
Definition. Rouché-Capelli’s theorem. Rouché-Capelli’s method to solve a linear system.

9. Gaussian elimination and how to apply the Gaussian elimination
Elementary operation. Determinant and rank computations.

10. Geometric vectors
Vectors in 2 dimensions. Sum and scalar product of vectors. Vectors in 3 dimensions. Linear combination. Linear dependence and independence of vectors.

11. Vector spaces and Linear subspaces
Definition, examples and properties.

12. How to generate linear subspaces
Linear combination and generators.

13. Linear dependence and independence

14. Basis of a linear subspace
Basis. Dimensions. Solution set of an homogeneous system and its dimensions.

15. Vector subspaces sum and intersection
Grassmann formula.

16. Affine space
Lines in planes and space. Affine subspace.

17. Homomorphisms
Homomorphism between vector spaces. Represent an homomorphism as a matrix.

18. Image of a matrix
Properties of an homomorphism image and how to compute it. Surjective homomorphism.

19. Kernel of a matrix
Properties of an homomorphism kernel and how to compute it. Injectivity homomorphism.

20. Endomorphisms
Matrix representation of the endomorphisms. Change of basis.

21. Eigenvalues and Eigenvectors
Definition and properties. Eigenspaces. Characteristic polynomial. Diagonalizable matrix.

22. Diagonalizing a matrix
Rules and examples.

Core Documentation

G. Accascina e V. Monti,
"Geometria"

Reference Bibliography

Giulia Maria Piacentini Cattaneo "Matematica discreta" Edito da Zanichelli.

Type of delivery of the course

Teaching includes lectures, exercises with the teacher and self-assessment tests for students. Attendance is not mandatory but is strongly recommended.

Attendance

Teaching includes lectures, exercises with the teacher and self-assessment tests for students. Attendance is not mandatory but is strongly recommended.

Type of evaluation

intermediate / final, oral / written

Programme

1. Linear equations and numbers
Systems of linear equations. Represent linear systems with matrix equations. Equivalent systems. Natural, integers, rational, real numbers and their properties. Pills of Graph Theory: subsets and set subtraction.

2. Matrixes and sets
Matrix with real coefficients. Square, triangular, diagonal matrixes. Transposed and symmetric matrix. Pills of Graph Theory: union and intersection.

3. Vector spaces and matrices
Matrix addition and scalar product and their properties.

4. Matrix product
Matrix product and its mathematical properties: associative and distributive. Example of how matrix product does not satisfy the commutative and simplification properties. Matrices and linear systems

5. Determinants
Formal definition of determinant and its properties. Transposed and triangular matrix determinants. Binet’s theorem.

6. Inverse of a matrix
Identity matrix. Inverse of a matrix and its properties. Cramer’s theorem.

7. Matrix rank
Definition and properties. Minor of a matrix.

8. System of linear equations.
Definition. Rouché-Capelli’s theorem. Rouché-Capelli’s method to solve a linear system.

9. Gaussian elimination and how to apply the Gaussian elimination
Elementary operation. Determinant and rank computations.

10. Geometric vectors
Vectors in 2 dimensions. Sum and scalar product of vectors. Vectors in 3 dimensions. Linear combination. Linear dependence and independence of vectors.

11. Vector spaces and Linear subspaces
Definition, examples and properties.

12. How to generate linear subspaces
Linear combination and generators.

13. Linear dependence and independence

14. Basis of a linear subspace
Basis. Dimensions. Solution set of an homogeneous system and its dimensions.

15. Vector subspaces sum and intersection
Grassmann formula.

16. Affine space
Lines in planes and space. Affine subspace.

17. Homomorphisms
Homomorphism between vector spaces. Represent an homomorphism as a matrix.

18. Image of a matrix
Properties of an homomorphism image and how to compute it. Surjective homomorphism.

19. Kernel of a matrix
Properties of an homomorphism kernel and how to compute it. Injectivity homomorphism.

20. Endomorphisms
Matrix representation of the endomorphisms. Change of basis.

21. Eigenvalues and Eigenvectors
Definition and properties. Eigenspaces. Characteristic polynomial. Diagonalizable matrix.

22. Diagonalizing a matrix
Rules and examples.

Core Documentation

G. Accascina e V. Monti,
"Geometria"

Reference Bibliography

Giulia Maria Piacentini Cattaneo "Matematica discreta" Edito da Zanichelli.

Type of delivery of the course

Teaching includes lectures, exercises with the teacher and self-assessment tests for students. Attendance is not mandatory but is strongly recommended.

Attendance

Teaching includes lectures, exercises with the teacher and self-assessment tests for students. Attendance is not mandatory but is strongly recommended.

Type of evaluation

intermediate / final, oral / written

Programme

Elements of set theory. Maps between sets: invective, surjective, bijective maps.
Elements of propositional logic, truth tables. Equivalence and order relations.
Combinatorics. Binomial coefficients and binomial theorem. Permutations. The Integers: divisibility, GCD and Euclidean algorithm, Bézout identity, linear congruences.
Baiscs of algebraic structures: permutation groups, abstract groups, polynomials and finite fields.
Elements of graph theory.Lattices and Boolean algebras

Core Documentation

Giulia Maria Piacentini Cattaneo
Matematica discreta e applicazioni
Zanichelli 2008

Reference Bibliography

Giulia Maria Piacentini Cattaneo "Matematica discreta" Edito da Zanichelli.

Type of delivery of the course

lectures

Attendance

recommended attendance

Type of evaluation

written test

Programme

Elements of Set Theory.
Union, intersection, Cartesian product, set subtraction, complementary set, cardinality.

Set functions.
Domain, codomain, Range. Injective, surjective, bijective functions. Inverse function, Identity, Permutations.

Elements of logic.
Propositional calculus, Operations between propositions.

Relations.
Reflexive, symmetric, antisymmetric, transitive. Order and equivalence relations. Examples.
Equivalence relations and classes. Quotient set.

Integer numbers, Division and its properties.
Greatest common divisor. Euclidean algorithm. Prime numbers. Fundamental theorem of arithmetic.

Congruence mod n.
Basic modular arithmetic. Sum and multiplication in Zn. Linear congruence. Description of linear congruence solutions. Euler's totient function. Small Fermat Theorem, Euler’s Theorem

Combinatory algebra.
Dispositions and combinations with and without repetitions, binomial coefficient. Properties. Tartaglia's triangle.

Partially ordered sets
Hasse Diagram. Maximum and minimum, Sup and inf. Reticular formations. Properties of inf e sup. Boolean Algebra. The Boolean operators AND, OR and NOT.

Core Documentation

Giulia Maria Piacentini Cattaneo
"Matematica discreta"
Edito da Zanichelli.

Reference Bibliography

Giulia Maria Piacentini Cattaneo "Matematica discreta" Edito da Zanichelli.

Type of delivery of the course

Classroom Teaching

Attendance

Not mandatory

Type of evaluation

A written test with questions regarding the theory and exercises, plus a possible oral test.

Programme

Lattices, Boolean algebras

Core Documentation

Giulia Maria Piacentini Cattaneo
Matematica discreta e applicazioni
Zanichelli 2008

Reference Bibliography

Giulia Maria Piacentini Cattaneo "Matematica discreta" Edito da Zanichelli.

Type of delivery of the course

lectures

Attendance

recommended attendance

Type of evaluation

written test

Programme

1. Linear equations and numbers
Systems of linear equations. Represent linear systems with matrix equations. Equivalent systems. Natural, integers, rational, real numbers and their properties. Pills of Graph Theory: subsets and set subtraction.

2. Matrixes and sets
Matrix with real coefficients. Square, triangular, diagonal matrixes. Transposed and symmetric matrix. Pills of Graph Theory: union and intersection.

3. Vector spaces and matrices
Matrix addition and scalar product and their properties.

4. Matrix product
Matrix product and its mathematical properties: associative and distributive. Example of how matrix product does not satisfy the commutative and simplification properties. Matrices and linear systems

5. Determinants
Formal definition of determinant and its properties. Transposed and triangular matrix determinants. Binet’s theorem.

6. Inverse of a matrix
Identity matrix. Inverse of a matrix and its properties. Cramer’s theorem.

7. Matrix rank
Definition and properties. Minor of a matrix.

8. System of linear equations.
Definition. Rouché-Capelli’s theorem. Rouché-Capelli’s method to solve a linear system.

9. Gaussian elimination and how to apply the Gaussian elimination
Elementary operation. Determinant and rank computations.

10. Geometric vectors
Vectors in 2 dimensions. Sum and scalar product of vectors. Vectors in 3 dimensions. Linear combination. Linear dependence and independence of vectors.

11. Vector spaces and Linear subspaces
Definition, examples and properties.

12. How to generate linear subspaces
Linear combination and generators.

13. Linear dependence and independence

14. Basis of a linear subspace
Basis. Dimensions. Solution set of an homogeneous system and its dimensions.

15. Vector subspaces sum and intersection
Grassmann formula.

16. Affine space
Lines in planes and space. Affine subspace.

17. Homomorphisms
Homomorphism between vector spaces. Represent an homomorphism as a matrix.

18. Image of a matrix
Properties of an homomorphism image and how to compute it. Surjective homomorphism.

19. Kernel of a matrix
Properties of an homomorphism kernel and how to compute it. Injectivity homomorphism.

20. Endomorphisms
Matrix representation of the endomorphisms. Change of basis.

21. Eigenvalues and Eigenvectors
Definition and properties. Eigenspaces. Characteristic polynomial. Diagonalizable matrix.

22. Diagonalizing a matrix
Rules and examples.

Core Documentation

G. Accascina e V. Monti,
"Geometria"

Reference Bibliography

Giulia Maria Piacentini Cattaneo "Matematica discreta" Edito da Zanichelli.

Type of delivery of the course

Teaching includes lectures, exercises with the teacher and self-assessment tests for students. Attendance is not mandatory but is strongly recommended.

Attendance

Teaching includes lectures, exercises with the teacher and self-assessment tests for students. Attendance is not mandatory but is strongly recommended.

Type of evaluation

intermediate / final, oral / written

Programme

1. Linear equations and numbers
Systems of linear equations. Represent linear systems with matrix equations. Equivalent systems. Natural, integers, rational, real numbers and their properties. Pills of Graph Theory: subsets and set subtraction.

2. Matrixes and sets
Matrix with real coefficients. Square, triangular, diagonal matrixes. Transposed and symmetric matrix. Pills of Graph Theory: union and intersection.

3. Vector spaces and matrices
Matrix addition and scalar product and their properties.

4. Matrix product
Matrix product and its mathematical properties: associative and distributive. Example of how matrix product does not satisfy the commutative and simplification properties. Matrices and linear systems

5. Determinants
Formal definition of determinant and its properties. Transposed and triangular matrix determinants. Binet’s theorem.

6. Inverse of a matrix
Identity matrix. Inverse of a matrix and its properties. Cramer’s theorem.

7. Matrix rank
Definition and properties. Minor of a matrix.

8. System of linear equations.
Definition. Rouché-Capelli’s theorem. Rouché-Capelli’s method to solve a linear system.

9. Gaussian elimination and how to apply the Gaussian elimination
Elementary operation. Determinant and rank computations.

10. Geometric vectors
Vectors in 2 dimensions. Sum and scalar product of vectors. Vectors in 3 dimensions. Linear combination. Linear dependence and independence of vectors.

11. Vector spaces and Linear subspaces
Definition, examples and properties.

12. How to generate linear subspaces
Linear combination and generators.

13. Linear dependence and independence

14. Basis of a linear subspace
Basis. Dimensions. Solution set of an homogeneous system and its dimensions.

15. Vector subspaces sum and intersection
Grassmann formula.

16. Affine space
Lines in planes and space. Affine subspace.

17. Homomorphisms
Homomorphism between vector spaces. Represent an homomorphism as a matrix.

18. Image of a matrix
Properties of an homomorphism image and how to compute it. Surjective homomorphism.

19. Kernel of a matrix
Properties of an homomorphism kernel and how to compute it. Injectivity homomorphism.

20. Endomorphisms
Matrix representation of the endomorphisms. Change of basis.

21. Eigenvalues and Eigenvectors
Definition and properties. Eigenspaces. Characteristic polynomial. Diagonalizable matrix.

22. Diagonalizing a matrix
Rules and examples.

Core Documentation

G. Accascina e V. Monti,
"Geometria"

Reference Bibliography

Giulia Maria Piacentini Cattaneo "Matematica discreta" Edito da Zanichelli.

Type of delivery of the course

Teaching includes lectures, exercises with the teacher and self-assessment tests for students. Attendance is not mandatory but is strongly recommended.

Attendance

Teaching includes lectures, exercises with the teacher and self-assessment tests for students. Attendance is not mandatory but is strongly recommended.

Type of evaluation

intermediate / final, oral / written