The course aims to provide an introduction to those aspects of linear and discrete mathematics needed in science and engineering.
Curriculum
Canali
teacher profile teaching materials
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
Matematica discreta e applicazioni
Zanichelli 2008
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 CattaneoMatematica discreta e applicazioni
Zanichelli 2008
Reference Bibliography
noneType of delivery of the course
lecturesAttendance
recommended attendanceType of evaluation
written test teacher profile teaching materials
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.
Programme
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.
Core Documentation
Piacentini Cattaneo, Matematica discreta. Zanichelli.
Delizia-Longobardi-Maj-Nicotera, Matematica Discreta, McGraw Hill.
Procesi-Rota, Elementi di algebra e matematica discreta. Accademica.
Type of evaluation
two hours written exam with theorical and partical problemsCanali
teacher profile teaching materials
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
Matematica discreta e applicazioni
Zanichelli 2008
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 CattaneoMatematica discreta e applicazioni
Zanichelli 2008
Reference Bibliography
noneType of delivery of the course
lecturesAttendance
recommended attendanceType of evaluation
written test teacher profile teaching materials
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.
Programme
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.
Core Documentation
Piacentini Cattaneo, Matematica discreta. Zanichelli.
Delizia-Longobardi-Maj-Nicotera, Matematica Discreta, McGraw Hill.
Procesi-Rota, Elementi di algebra e matematica discreta. Accademica.
Type of evaluation
two hours written exam with theorical and partical problems