Let X be a set. Let P be a set of subsets of X such that: • ØEP; • the union of all sets A E P is X. Note that these are clauses (a) and (c) of the definition of a partition Now define a relation R on the set X by R = {(x, y) :x EA and y E A for some A EP}, as in Theorem 1.7(b). Which of the following is true? Select one: O a. R must be symmetric and transitive but might not be reflexive. O b. R must be reflexive and symmetric but might not be transitive. C. R must be an equivalence relation, and { [X]R:X EX} must equal P. O d. R must be reflexive and transitive but might not be symmetric. e. R must be an equivalence relation, but { [x]R:X EX} might not be equal to P.

Let X be a set. Let P be a set of subsets of X such that: • ØEP; • the union of all sets A E P is X. Note that these are clauses (a) and (c) of the definition of a partition Now define a relation R on the set X by R = {(x, y) :x EA and y E A for some A EP}, as in Theorem 1.7(b). Which of the following is true? Select one: O a. R must be symmetric and transitive but might not be reflexive. O b. R must be reflexive and symmetric but might not be transitive. C. R must be an equivalence relation, and { [X]R:X EX} must equal P. O d. R must be reflexive and transitive but might not be symmetric. e. R must be an equivalence relation, but { [x]R:X EX} might not be equal to P.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 6TFE: Label each of the following statements as either true or false. Let R be a relation on a nonempty...

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