Menú

# MAY

## Monday, May 30th, 2011.

• Graphs - Lecture 14. Digraphs. Oriented paths, cycles. Acyclic digraphs. Ordering induced by an acyclic graph. Review of partial and total orderings. Total orderings compatible with partial orderings. Topological sorting. Existence of minimal elements in finite partial relations. Download this video . "Size:2.05 GB" .

## Thursday, May 26th, 2011.

• Characterization of trees. Rooted trees. Root, internal nodes, leaves. Relatives: father, siblings, ancestors, descendant. Isomorphic trees.
• Problem set: exercise 31.

## Monday, May 23th, 2011.

• Graphs - Lecture 8. Connectivity in multigraphs: Eulerian circuits, Eulerian multigraphs, Eulerian paths, semi-Eulerian multigraphs. Sufficient and necessary conditions for a multigraph to be Eulerian. Sufficient conditions for a multigraph to be semi-Eulerian. Download this video . "Size:1.57 GB" .

## Tuesday, May 17th, 2011.

• Graphs - Lecture 7. Connectivity in simple graphs: paths, simple paths, circuits, simple circuits. Connectivity in directed graphs. Connectivity between pair of vertices. Cut vertices and cut edges. Download this video . "Size:974 MB" .

## Tuesday, May 10th, 2011.

• Graphs - Lecture 2. Modeling with graphs (continued): acquaintanceship graphs and call graphs. Basic terminology: adjacency, incidence, connectivity. Degree of a vertex in an undirected graph. The handshaking theorem and related results. Degree of a vertex in a directed graph. The handshaking theorem and related results for directed graphs.Download this video . "Size:1.68GB" .

## Monday, May 9th, 2011.

• Graphs - Lecture 1. Introduction to graph theory. Definition of a graph. Vertex set, edge set. Types of graphs: simple graphs, multigraphs, pseudographs, directed graphs. Examples of those graphs. Modeling with graphs. Precedence graphs.Download this video . "Size:1.37GB" .

## Wednesday, May 4th, 2011.

There was a review class and it was not recorded.

# APRIL

## Wednesday, April 27th, 2011.

• Induction and Recursion - Lecture 9. Problem set: exercises 9 and 11. Review of lists: definition of a list, length of a list, rest of a list, head of a list, concatenation of lists. Structural tree of a list. Depth of a list.Download this video . "Size:1.5GB" .

## Tuesday, April 26th, 2011.

• Propositional Logic.- Lecture 19. Problem set: exercise 18 (f).Download this video . "Size:605 MB" Warning: due to technical reasons, some parts of this video are not properly recorded. We apologize for the inconveniences.

## Tuesday, April 12th, 2011.

• Induction and Recursion.- Lecture 6. Dependency trees and initial examples. Types of recursive functions to be dealt with. Formal definition of dependency trees. More examples.Download this video . "Size:1.29 GB" .

## Monday, April 11, 2011.

• Induction and Recursion.- Lecture 5. Recursive definition of sets. Examples: numerical sets, lists. Formal definition of lists. Functions on lists: LONG, REST, CAB, the set of LIST_p.Download this video . "Size:3.01 GB" .

## Friday, April 8th, 2011.

• Induction and Recursion - Lecture 3 and Lecture 4. Recursive definitions of sets, sequences and maps/functions. Examples. Recursive correspondences. Well-defined recursive maps. Fibonacci numbers. More examples of induction proofs. Errors in inductive proofs. Examples.Download this video . "Size: GB" .

## Wednesday, April 6th, 2011.

• Induction and recursion - Lecture 2. Well-Ordering Property. Proof of the equivalence between the Well-Ordering Property and the Induction Principle. Other forms of induction. Strong induction. Examples.Download this video . "Size: 642MB" .

## Tuesday, April 5th, 2011.

• Induction and recursion - Lecture 1. Properties depending on natural numbers. Inductive proofs: basis step and inductive step. Examples: summation and inequalities.Download this video . "Size: 2.35GB"
• Propositional Logic - Lecture 13. Problem 15

## Friday, April 1st, 2011.

• Propositional Logic -. Lecture 10. Replacement theorem. Further elaboration of tableaux rules (implication and biconditional). More on closed trees and unsatisfiable formulae. Deductive argument: hypotheses and conclusion. Correct and incorrect deductive arguments. Relation between argument correctness and unsatisfiable formulae.Download this video . "Size: 2.35GB" .

# MARCH

## Wednesday, March 30th, 2011.

• Lecture 8. Problem set: exercises 7 to 10. Sets of formulae. Generalization of the satisfiable formula, tautology and contradiction to sets of formulae. Download this video . "Size: 2.32GB" .

## Tuesday, March 29th, 2011.

• Lecture 7. Analytic tableaux. Simplifiable, developable, disjunctive and conjunctive formulae. Description of the construction of an analytic tableaux. Halting conditions. Obtaining the models from the tree. Determining whether a formula is tautology or contradiction. Examples. Download this video . "Size: 2.32GB" .

## Monday, March 28th, 2011.

• Lecture 6. Logical equivalences as a equivalence relation. The class of the constant true and constant false. Results on equivalence relations. Download this video . "Size: 2.32GB" .
• Lecture 6. Logical equivalences as a equivalence relation. The class of the constant true and constant false. Results on equivalence relations. Download this video . "Size: 2.32GB" .

## Friday, March 24th, 2011.

• Propositional Logic -. Lecture 4. Semantics. Valuations. Meaningful valuations. Models of formulae. Non-models. Tautologies, contradictions and contingencies. Download this video . "Size: 2.32GB" .
• Propositional Logic -. Lecture 5. Satisfiable formulae. Unsatisfiable formulae. Results relating tautology and satisfiable formulae, and contradiction and unsatisfiable formulae. Download this video . "Size: 964MB" .

## Wednesday, March 23th, 2011.

• Propositional Logic - Lecture 3. Structural Recursion Principle (continued). Several examples including functions that transform formulae into formulae. Structural tree. Examples.Download this video . "Size: 2.07GB" .

## Tuesday, March 22nd, 2011.

• Propositional Logic -. Lecture 2. Logic Syntax. Primitive symbols. Propositional variables. Logical constants. Literal. Main connective. Formation rules for formulae. Order of precedence. Formula simplification. Structural Recursion Principle. Examples.Download this video . "Size: 2.19GB" .

## Monday, March 21st, 2011.

• Propositional Logic -. The purpose of logic. Propositions. Declarative sentences. Formal languages. Formalization. Simple and compound statements. Connectives: negation, disjunction, exclusive or, conjunction, implication and biconditional connectives. Table truth of connectives. Converse, contrapositive and inverse of an implication. Translating English sentences to a formal language. Necessary and sufficient conditions.Download this video . "Size: 1.8GB" .

## Friday, March 11th, 2011.

• Combinatorics 9 -. Discrete probability. Random experiments. Outcomes. Sample space. Probability functions. Elementary events. Event operations in terms of set operations. Properties of events. Laplace's rule. Examples. Problem set: problems 15 - 17 Download this video . "Size: 2.26GB" .

## Wednesday, March 9th, 2011.

• Combinatorics 8 -. Problems on permutation with repetitions, r-permutations with repetitions and combinations with repetitions. Counting the number of natural solutions of a linear equation. Problems 10-14. Download this video. "Size: 2.20GB"

## Tuesday, March 8th, 2011.

• Combinatorics 7 -. Review of counting techniques. Counting with repetitions. r-Permutations with repetitions. Permutations with repetitions. Solving an integer equation. Combinations with repetitions. Download this video. "Size: 1.70GB"

## Monday, March 7th, 2011.

• Combinatorics 6 -. The damned problem 9, solved in two ways: by using the complement and by computing directly the desired cardinality. Binomial Theorem. Proof of the Binomial Theorem. Properties that can be proved through the Binomial Theorem. Download this video. "Size: 1.76GB"

## Tuesday, March 1st, 2011.

• Combinatorics 3 - The Pigeon Principle. The Generalized Pigeon Principle. Ordered selections of elements. Permutations. r-Permutations. Examples Download this video. "Size: 1.8GB"

# FEBRUARY

## Friday, February 25th, 2011.

• Combinatorics - Class 1: the art of counting, sum rule, product rule, general strategy to solve combinatorics problems. Examples: counting divisors, bit strings, maps, injective maps Download this video. "Size: 1.78GB"

## Tuesday, February 22nd, 2011.

• Order Relations - 4. Ordering Relations: posets, comparable elements, total ordering, Hasse diagrams, maximal elements, minimal elements, maximum element, and minimum element. Download this video. Size: 1.26GB.

## Monday, February 21st, 2011.

• Equivalence Relations and Partitions - 3. Equivalence relations, Theorem 1 (on page 560) on how an equivalence relation induces a partition, definition of partition, relation between equivalence relations and partitions. Problems on relations. Download this video.

## Friday, February 18th, 2011.

The class consisted of group work and there was no recording of the class that day.

## Tuesday, February 15th, 2011.

• Relations class 1: general definition of relation. Graphical representation of relations: sagittal diagrams. Relations on a set. Reflexive, symmetric, antisymmetric and transitive relations. Equivalence and order relations. Download this video.
• Explanation Magic Prediction by Piling. Clarification of some points about the project Magic Prediction by Piling. Download this video.

## Monday, February 14th, 2011.

• Functions class 2: invertible functions, function composition, the floor function. Download this video.

## Friday, February 10th, 2011.

• Sets and Functions - 3. Set operations: union, intersection, complement, difference. Functions: correspondence, maps, functions; injective, surjective and bijective functions. Download this video.

## Wednesday, February 9th, 2011.

• English Phonectics 3. Review of vowels, consonants, semi-consonants, pronunciation of plurals, pronunciation of possessives, pronunciation of the third person of singular of Present Simple.
• Sets - 2: power set, number of elements of the power set, questions from the problem set. Download this video.

## Tuesday, February 8th, 2011.

• English Phonetics - 2.
• Sets - 1: definition of a set, set membership, set equality, set inclusion, important sets. Download this video.

## Monday, February 7th, 2011.

### Docencia

• ##### Proyectos de fin de carrera 