Exam IV Study Sheet

MA:221, Fall 2012

All material from previous exams (concepts not tested explicitly, but are necessary) Definitions/Terms

reflexivesymmetricantisymmetrictransitiveposet

partial orderingdirected graphundirected graphHasse diagram upper boundlower bound least upper bound greatest lower bound latticetotal order equivalence relationequivalence class partitionpart/cell/block alphabet stringempty string string lengthconcatenation prefix/suffixsubstring language Chapter 7 – More on Relations

Properties: reflexivity, symmetry, antisymmetry, transitivity – know definitions and understand what they mean about a relation, know and be able to construct examples of relations with and without each property (or combination of properties). Be able to count reflexive, symmetric, and antisymmetric relations. Be able to prove whether a described relation has or does not have each of these properties. Representations of relations: as sets, as matrices, and as digraphs. Understand the relationship among these representations. Be able to form compositions of relations, and understand how the notion of composition translates to matrices and digraphs. Understand how relation properties are manifested in each of these representations. Partially ordered sets: definitions, examples, be able to prove that a relation is a partial ordering, be able to construct and analyze Hasse diagrams, be able to spot upper bounds, lower bounds, the greatest lower bound, and least upper bound for a given set. Equivalence Relations: definitions, examples, equivalence classes, partitions, and the relationship between partitions and equivalence relations, be able to construct the partition induced by a given equivalence relation, and the equivalence relation arising from a given partition.

Chapter 6 – Strings

String theory: definitions, alphabets, powers of an alphabet ((0, (n, (*, (+, etc.),...

MA:221, Fall 2012

All material from previous exams (concepts not tested explicitly, but are necessary) Definitions/Terms

reflexivesymmetricantisymmetrictransitiveposet

partial orderingdirected graphundirected graphHasse diagram upper boundlower bound least upper bound greatest lower bound latticetotal order equivalence relationequivalence class partitionpart/cell/block alphabet stringempty string string lengthconcatenation prefix/suffixsubstring language Chapter 7 – More on Relations

Properties: reflexivity, symmetry, antisymmetry, transitivity – know definitions and understand what they mean about a relation, know and be able to construct examples of relations with and without each property (or combination of properties). Be able to count reflexive, symmetric, and antisymmetric relations. Be able to prove whether a described relation has or does not have each of these properties. Representations of relations: as sets, as matrices, and as digraphs. Understand the relationship among these representations. Be able to form compositions of relations, and understand how the notion of composition translates to matrices and digraphs. Understand how relation properties are manifested in each of these representations. Partially ordered sets: definitions, examples, be able to prove that a relation is a partial ordering, be able to construct and analyze Hasse diagrams, be able to spot upper bounds, lower bounds, the greatest lower bound, and least upper bound for a given set. Equivalence Relations: definitions, examples, equivalence classes, partitions, and the relationship between partitions and equivalence relations, be able to construct the partition induced by a given equivalence relation, and the equivalence relation arising from a given partition.

Chapter 6 – Strings

String theory: definitions, alphabets, powers of an alphabet ((0, (n, (*, (+, etc.),...

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