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(d) There exists a relation U on A so that U is symmetric and antisymmetric, but U is not transitive.

(d) There exists a relation U on A so that U is symmetric and antisymmetric, but U is not transitive.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 14E: In each of the following parts, a relation is defined on the set of all human beings. Determine...
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How do you deal with part(d)?I need detailed solution for it.Thank you!
1. Let A = {1, 2, 3, 4}. Prove or disprove each of the following statements. If the statement is true, you must simply describe the relation you choose as a subset of A x A and draw its directed graph (arrow diagram). If the statement is false, then you must explain (prove) why there exists no such relations. (a) There exists a relation R on A so that R is reflexive and symmetric, but R is not antisymmetric nor transitive. (b) There exists a relation S on A so that S is symmetric and transitive, but S is not antisymmetric nor reflexive. (c) There exists a relation T on A so that T is antisymmetric and transitive, but T is not symmetric nor reflexive. (d) There exists a relation U on A so that U is symmetric and antisymmetric, but U is not transitive.
Transcribed Image Text:1. Let A = {1, 2, 3, 4}. Prove or disprove each of the following statements. If the statement is true, you must simply describe the relation you choose as a subset of A x A and draw its directed graph (arrow diagram). If the statement is false, then you must explain (prove) why there exists no such relations. (a) There exists a relation R on A so that R is reflexive and symmetric, but R is not antisymmetric nor transitive. (b) There exists a relation S on A so that S is symmetric and transitive, but S is not antisymmetric nor reflexive. (c) There exists a relation T on A so that T is antisymmetric and transitive, but T is not symmetric nor reflexive. (d) There exists a relation U on A so that U is symmetric and antisymmetric, but U is not transitive.
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