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Theorem. Let < be a preorder on a set X. Define the relation =, where x = y holds if and only if x < y and y < x. Then = is an equivalence relation on X.

Theorem. Let < be a preorder on a set X. Define the relation =, where x = y holds if and only if x < y and y < x. Then = is an equivalence relation on X.

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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A binary relation < on a set X is said to be a preorder if it is reflexive and transitive. For example, let X denote the collection of all living earthlings and let < be defined as follows: V x, y € X, x < y if and only if x is no older than y, where age is measured in whole days. Clearly < is reflexive and transitive, but it is not symmetric (since younger doesn't imply older) nor antisymmetric (since different people may be the same age). But prove the following. Theorem. Let < be a preorder on a set X. Define the relation =, where x = y holds if and only if x <y and y < x. Then = is an equivalence relation on X.
Transcribed Image Text:A binary relation < on a set X is said to be a preorder if it is reflexive and transitive. For example, let X denote the collection of all living earthlings and let < be defined as follows: V x, y € X, x < y if and only if x is no older than y, where age is measured in whole days. Clearly < is reflexive and transitive, but it is not symmetric (since younger doesn't imply older) nor antisymmetric (since different people may be the same age). But prove the following. Theorem. Let < be a preorder on a set X. Define the relation =, where x = y holds if and only if x <y and y < x. Then = is an equivalence relation on X.
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