Università Roma Tre

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

Giulia Maria Piacentini Cattaneo
Matematica discreta e applicazioni
Zanichelli 2008

Basics of set theory. Union, intersection, Cartesian product,
difference, complement. Set of parts of a finite whole, and its
cardinality.

Elements of logic: propositional calculus. Negation operations,
conjunction, disjunction, XOR, logical implication, double implication.
Truth tables. Logical equivalence. Tautologies and contradictions. hints
on predicates. Universal and existential quantifier.

Applications between sets. Domain, co-domain, image, counter-image.
Injective, surjective, bijective applications. Reverse application.
Operating product between applications. Identity. The set of applications
between two finite sets and its cardinality. Permutations.

Relations. Reflexive, symmetric, antisymmetric, transitive property:
order and equivalence relation. Examples of relationships. Partially together
ordered. Equivalence relations, equivalence classes, set
quotient.

Integers: divisibility and its properties. Division with the remainder. Maximum
common divider. Euclid's algorithm. Identity of Bézout, algorithm
extended Euclid's. Diophantine equations. Application of the algorithm
Euclid looking for integer solutions for the equation ax + by = c. Numbers
first. Fundamental theorem of arithmetic and Euclid's theorem.

Consistency form no. The set Z/nZ of the remainder classes modulo n. Sum and
multiplication in Z/nZ. Linear congruences. Condition for resolvability.
Description of the solutions of linear congruences. Congruence systems
and the Chinese remainder theorem. Invertible elements in Z/nZ. Euler's φ function.
Fermat's little theorem, Euler-Fermat theorem.

Combinatorics: Arrangements and combinations without repetitions, coefficients
binomials. Properties of binomial coefficients, Development of the binomial.
Arrangements and combinations with repetitions, Tartaglia triangle.

Partially ordered sets, Hasse diagrams. maximum and minimum,
maximal and minimal elements, higher and lower, upper and lower.
lattices. Properties of inf and sup in a lattice. Algebraic lattices. lattices
limited, complementary, distributive. Boolean algebras.

Piacentini Cattaneo, Matematica discreta. Zanichelli.
Delizia-Longobardi-Maj-Nicotera, Matematica Discreta, McGraw Hill.
Procesi-Rota, Elementi di algebra e matematica discreta. Accademica.

1. Linear equations and numbers
Linear equations systems. Matrix associated with a linear system. Equivalent systems. Natural, integer, rational numbers, real numbers and their property. Recall of set theory: inclusion of sets, difference between sets.
2. Matrices and sets
Matrices with real coefficients. Square, triangular, diagonal matrices. Transpose of a matrix and symmetric matrices. Recall of set theory: union and intersection of sets.
3. The vector space of the matrices
Addition between matrices and its properties. Multiplying a scalar for a matrix and its properties.
4. Product between matrices
Product between matrices with compatible dimensions. Properties of the product: associative property and distributive property. Examples showing that product between matrices does not satisfy the commutative property and the simplification property. Matrices and linear systems.
5. Determinants
Definition by induction of the determinant when using the first-row development. Determinant property: development according to any row or column, determinant of the transposed matrix, determinant of a triangular matrix. Binet theorem.
6. Reverse matrix
Unit matrix. Reverse matrix. Inverse property. Cramer's theorem.
7. Rank of a matrix
Definition. Property of the rank. Minors of a matrix. Theorem of Kronecker.
8. Linear equation systems
Definitions. Rouché-Capelli theorem. Rouché-Capelli method for solving a linear system.
9. Gauss method
10. Applications of Gauss method
Basic operations. Calculation of the determinant. Calculation of the rank.
11. Geometric vectors
Plan vectors. Addition between vectors. Product of a vector for a scalar. Space vectors. Lines and planes for the origin. Average point.
12. Linear combinations of geometric vectors
Linear combinations. Linearly dependent and independent vectors. Characterization of linearly independent vectors in V2(O) and V3(O).
13. Vector spaces on the real numbers
Definition of vector spaces. Examples of vector spaces. Basic properties of vector spaces.
14. Vector subspaces
Definition of vector spaces. Subspaces of V2(O) and V3(O).
15. Generators of vector spaces
Linear combinations and generators.
16. Linear dependency and independency
17. Basis of vector spaces
Basis. Dimension. Dimension of the set of solutions of a homogeneous system. Dimension of subspaces. Calculation of dimensions and basis.
18. Intersection and sum of subspaces
Intersection of vector subspaces. Sum of vector subspaces. Grassmann's formula.
19. Affine subspaces
The lines of the plane and of the space. Space planes. Affine subspaces. Set of system solutions.
20. Homomorphisms
Homomorphisms between vector spaces. Matrix associated with a homomorphism. Homomorphism associated with a matrix.
21. Image
Property of the image of a homomorphism. Calculation of the image of a homomorphism. Condition of surjectivity of a homomorphism.
22. Kernel
Property of the kernel of a homomorphism. Calculation of the kernel of a homomorphism. Injectivity condition of a homomorphism.
23. Endomorphisms
Matrix associated with an endomorphism. Change of basis.
24. Eigenvalues ​​and eigenvectors
Definitions and basic properties. Eigenspaces. Characteristic polynomial. Diagonalizable matrices.
25. Diagonalization
Diagonalizability conditions. Diagonalization procedure.

G. Accascina and V. Monti, Geometry*
* This book is available for free at the following link: http://www.dmmm.uniroma1.it/accascinamonti/geogest/Geometria.pdf

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.

G. Accascina e V. Monti,
"Geometria"

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.

G. Accascina e V. Monti,
"Geometria"