离散数学

章节摘录

版权页：   插图：    We have seen how to construct a new relation by compositing two existing relations.Let's look at another way to construct a new relation from an existing relation. We willstart with a binary relation R and try to construct another relation containing R that alsosatisfies some particular properties. If R is a relation on a set A and P is a property, such as being reflexive, symmetric, ortransitive, then the P closure of R is the smallest binary relation that contains R andsatisfies property P. We denote the P closure of R by P (R). If R already satisfiesproperty P, then we have R = P(R). We will be concerned with three properties: reflexivity, symmetry, and transitivity. Thereflexive closure of R is denoted by r(R), the symmetric closure of R is denoted bys(R), and the transitive closure of R is denoted by t(R).Our goal is to find some techniques to compute these closures. We will start with arunning example to introduce the main idea.

书籍目录

1. Propositional Logic 1. 1 Propositions and Connectives 1. 2 Propositional WFF and Assignment 1. 3 Propositional Equivalences 1. 4 Disjunctive Normal Form 1.5 Functionally Complete Set of Logical Connectives 1.6 Rules of Inference 2. Predicate Logic 2.1 Predicates and Quantifiers 2.2 Well-Formed Formulas in Predicate Logic 2.3 Equivalent Formulas 2.4 Prenex Normal Form 2.5 Inference Rules in Predicate Calculus 3. Set Theory 3.1 Sets 3.2 Set Operations 3.3 Inclusion-Exclusion 4. Relations 4.1 Cartesian Products and Relations 4.2 Properties of Relations 4.3 Representing Relations 4.4 Closure of Relations 4.5 Equivalence Relations 4.6 Partial Orderings 5. Graphs 5.1 Graph Terminology 5.2 Representing Graphs and Graph Isomorphism 5.3 Subgraphs 5.4 Euler and Hamilton Paths 5.5 The Shortest-Path Problem 5.6 Planar Graphs 6. Trees 6.1 Basic Concepts 6.2 Roots and Orderings 6.3 Spanning Trees 7. Algebra Structures 7.1 Basic Concepts 7.2 Groups 7.3 Rings and Fields 7.4 Boolean Algebras Reference

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