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18. Operations on Relations

The chapter discusses operations on relations, including set-theoretic operations such as union, intersection, and composition. It explores the concepts of powers of relations and various closure properties, including reflexive, symmetric, and transitive closures. Through examples, it illustrates how relations can be manipulated and expanded to satisfy specific properties.

Sections

Operations on Relations

This section covers operations on relations in discrete mathematics, including union, intersection, and composition of relations.

18 Section Overview

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18.1 Union of Relations

This section discusses the operations on relations, focusing on union, intersection, and closure concepts in the context of discrete mathematics.

18.2 Intersection of Relations

This section explores set-theoretic operations on relations, including union, intersection, differences, powers, and closures.

18.3 Difference of Relations

This section explains the operations that can be performed on mathematical relations, specifically focusing on the difference, intersection, and union of relations.

18.4 Composition of Relations

This section discusses important operations on relations including set operations (union, intersection, difference) and introduces the concepts of composition of relations and their powers.

18.5 Powers of a Relation

This section discusses the powers of a relation and operations that can be performed on relations, including union, intersection, and composition.

18.6 Interpretation of Powers of Relation

This section discusses the operations on relations, focusing on the powers of a relation and how they can be interpreted in a graph context.

18.7 Closure of a Relation

This section explores the closure of a relation with respect to a certain property, detailing how to minimally expand a relation to satisfy given properties such as reflexivity, symmetry, and transitivity.

Reflexive Closure

This section discusses reflexive closure in relation to set operations, particularly focusing on how to ensure a relation includes all pairs of the form (a, a) for elements in a set.

18.7.1 Section Overview

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Symmetric Closure

This section discusses the concept of symmetric closure of a relation in discrete mathematics.

18.7.2 Section Overview

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Transitive Closure

This section introduces the concept of transitive closure of a relation and explains its significance in formal mathematics.

18.7.3 Section Overview

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Learning Objectives

  • Relations can undergo set-theoretic operations like union, intersection, and difference.

  • The composition of relations allows for establishing connections between different sets.

  • Closure of a relation refers to expanding it minimally to satisfy specified properties such as reflexivity and symmetry.

Key Concepts

Union of Relations

The union of two relations R1 and R2 includes all ordered pairs that are in either R1 or R2.

Composition of Relations

The composition of two relations R and S combines them such that if an element a is related to b through R, and b is related to c through S, then a is related to c in the composed relation.

Reflexive Closure

The reflexive closure of a relation R is formed by adding all pairs of the form (a, a) for each element a in the set if they are not already included in R.

Transitive Closure

The transitive closure of a relation R involves repeatedly adding pairs of the form (a, c) based on already existing pairs (a, b) and (b, c) until no new pairs can be added.

Practice Exercises

Total Questions

2

Estimated Time

4 min

Passing Score

70%

Instructions

  • Read each question carefully
  • You can use hints if you need help
  • Complete all questions before submitting
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