Let X be a set. Let P be a set of subsets of X such that:if A and B are distinct elements of P, then AnB = 0;the union of all sets A € Pis X.Note that these are clauses (b) and (c) of the definition of a partition (Definition 1.5).Now define a relation R on the set X by R = {(x, y) : x EA and ye A for some A € P}, as in Theorem 1.7(b). Which of the following is true?Select one:a. R must be an equivalence relation, but { [x]R: X EX} might not be equal to P.O b. R must be reflexive and transitive but might not be symmetric.O c. R must be symmetric and transitive but might not be reflexive.O d. R must be reflexive and symmetric but might not be transitive.O e. R must be an equivalence relation, and { [X]R: XEX} must equal P.

This is a question of Relation.

Let X be a set. Let P be a set of subsets of X such that: if A and B are distinct elements of P, then AnB = 0; the union of all sets A E P is X. Note that these are clauses (b) and (c) of the definition of a partition (Definition 1.5). Now define a relation R on the set X by R = {(x, y): xe A and ye A for some A € P}, as in Theorem 1.7(b). Which of the following is true? Select one: O a. R must be an equivalence relation, but { [x]R: X EX} might not be equal to P. O b. R must be reflexive and transitive but might not be symmetric. OC. R must be symmetric and transitive but might not be reflexive. O d. R must be reflexive and symmetric but might not be transitive.

Let X be a set. Let P be a set of subsets of X such that: if A and B are distinct elements of P, then AnB = 0; the union of all sets A E P is X. Note that these are clauses (b) and (c) of the definition of a partition (Definition 1.5). Now define a relation R on the set X by R = {(x, y): xe A and ye A for some A € P}, as in Theorem 1.7(b). Which of the following is true? Select one: O a. R must be an equivalence relation, but { [x]R: X EX} might not be equal to P. O b. R must be reflexive and transitive but might not be symmetric. OC. R must be symmetric and transitive but might not be reflexive. O d. R must be reflexive and symmetric but might not be transitive.

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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