Review: Discrete Mathematics and Its Applications

Contents Mathematical Logic Set Theory Counting Graph and Tree

Propositional Logic Propositions and Logical Operators Propositional Formula and Its Classification Tautology Contradiction Contingence

How to determine whether a compound proposition is a tautology (contradiction, contingence)? − truth table − logical equivalence − normal forms Propositional Equivalences − Propositions logically equivalent is an equivalent relation − Some important equivalences

Propositional Normal Forms − Conjunctive Clauses and Disjunctive Normal Form − Miniterm and Full Disjunctive Norm − Conjunctive Normal Form Valid of Argument − Valid of Argument − Rules of Inference

Predicate Logic Predicates and Quantifiers universe of discourse, banding Logical Equivalence − De Morgan’s laws for predicates − Quantifiers—-handle with care!

Inference for Quantified Statements But Still holds Still holds Inference for Quantified Statements

Set Theory Set and Set operations − Subsets, Proper Subsets − Set Operations and Set Identities Power Set

Cartesian Products ordered pair Cardinality of Finite sets − Principle of Inclusion-exclusion

Infinite Set Cardinality Infinite Set − cardinality − a infinite set has a proper subset with the same cardinality Countable Infinite Set − has the same cardinality with − Properties of the countable sets: The union of two countable sets, finite number of countable sets, countable number of countable sets are countable. − Some special infinite sets:

Uncountable Infinite Set − Let countable infinite set, uncountable − uncountable

Relation Relations and their representing − − Representing methods: Set, Matrix, Graph Properties of relations − Reflexive, Symmetric, Transitive Combining relations − Composite, Inverse

Closures of relations be a relation on set . − reflexive closure , where − symmetric closure − transitive closure Equivalence relation − Equivalence relation: reflexive, symmetric, transitive − Equivalence classes and their properties − Partition and equivalence relation

Partial ordering − Partial order: reflexive, antisymmetric and transitive. A set together with a partial ordering is called a partial order set (denoted by ), totally order − Hasse Diagram − maximal element, minimal element, greatest element, least element, upper bound, lower bound, least upper bound, greatest lower bound − Lattice: A partially ordered set in which pair of elements has both a least upper bound and a greatest lower bound

Counting Basic Counting Two basic counting principle − the sum rule − the product rule The Pigeonhole Principle Permutations and combinations − without repetition and with repetitions Generating permutations and combinations

Advanced Counting Techniques − Recurrence relations − Generating function − Solving recurrence relations、Counting with Generating function − Solving linear (non-)homogeneous recurrence relations with constant Characteristic Equation, Characteristic roots

Graphs and Trees Introduction to Graphs − Types of graphs: undirected [simple, multigraph, pseudograph], directed [directed graph, directed multigraph] − Some special simple graphs − Representing graphs and graph isomorphism − Connectedness Euler graph and Hamilton graph Planar Graph and Coloring − Euler formula, Chromatic number Shortest Path − Dijkstra’s algorithm

Tree Definition and Properties of Trees − tree: connected + no simple circuit − unique simple path between any two of its vertices − tree: vertices, edges − A full m-ary tree: internal vertices, vertices − If an m-ary tree of height has leaves, then . If the m-ary trees is full and balanced, then . Applications of Trees − Binary search trees − Prefix codes

Spanning Tree − Algorithms for constructing spanning trees: depth-first search (backtracking), breadth-first search. Minimal Spanning Tree − Algorithms for constructing minimal spanning trees: Kruskal’s algorithm, Prim’s algorithm