A relation R is called asymmetric if (a, b) ∈R implies that (b, a) ∉R . Exercises 18-24 explore

Name: Problem 22, Chapter 9: Discrete Mathematics and its Applications
Uploaded: 2020-07-28T02:27:01-08:00
Duration: 6 min 18 s
Channel: Chris Trentman
Description: A relation R is called asymmetric if (a, b) ∈R implies that (b, a) ∉R . Exercises 18-24 explore the notion of an asymmetric relation. Exercise 22 focuses on the difference between asymmetry and antisymmetry. Must an asymmetric relation also be antisymmetric? Must an antisymmetric relation be asymmetric? Give reasons for your answers.

step 1 We are given the definition of an asymmetric relation. We are asked some questions about the dif

Question

A relation $R$ is called asymmetric if $(a, b) \in R$ implies that $(b, a) \notin R .$ Exercises $18-24$ explore the notion of an asymmetric relation. Exercise 22 focuses on the difference between asymmetry and antisymmetry. Must an asymmetric relation also be antisymmetric? Must an antisymmetric relation be asymmetric? Give reasons for your answers.

   A relation $R$ is called asymmetric if $(a, b) \in R$ implies that $(b, a) \notin R .$ Exercises $18-24$ explore the notion of an asymmetric relation. Exercise 22 focuses on the difference between asymmetry and antisymmetry.
Must an asymmetric relation also be antisymmetric? Must an antisymmetric relation be asymmetric? Give reasons for your answers.

Discrete Mathematics and its Applications

Kenneth Rosen 8th Edition

Chapter 9, Problem 22

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Solved by Expert Chris Trentman

07/28/2020

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An antisymmetric relation is defined as a relation where if (a, b) is in R and (b, a) is in R, then a must be equal to b. Show more…

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A relation $R$ is called asymmetric if $(a, b) \in R$ implies that $(b, a) \notin R .$ Exercises $18-24$ explore the notion of an asymmetric relation. Exercise 22 focuses on the difference between asymmetry and antisymmetry. Must an asymmetric relation also be antisymmetric? Must an antisymmetric relation be asymmetric? Give reasons for your answers.

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Key Concepts

Asymmetric Relation

An asymmetric relation is one in which if (a, b) is in the relation, then (b, a) cannot be in the relation, for any distinct a and b. This excludes any possibility of symmetry; the relation is strictly one directional and inherently irreflexive, meaning no element can be related to itself under the definition of asymmetry.

Antisymmetric Relation

An antisymmetric relation is defined by the condition that if both (a, b) and (b, a) are in the relation, then a must equal b. This allows for the possibility that an element is related to itself, while ensuring that distinct elements cannot be mutually related in both directions. Thus, an antisymmetric relation can contain pairs where the reversed pair also exists if and only if the two elements are identical.

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