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Let X be a set. Let P be a set of subsets of X such that: • Ø & P; the union of all sets A € Pis X. Note that these are clauses (a) 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 reflexive and transitive but might not be symmetric. b. R must be reflexive and symmetric but might not be transitive. R must ho.cum

Let X be a set. Let P be a set of subsets of X such that: • Ø & P; the union of all sets A € Pis X. Note that these are clauses (a) 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 reflexive and transitive but might not be symmetric. b. R must be reflexive and symmetric but might not be transitive. R must ho.cum

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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QUESTION 8 Let X be a set. Let P be a set of subsets of X such that: • Ø & P; the union of all sets A E P is X. Note that these are clauses (a) 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 E P}, as in Theorem 1.7(b). Which of the following is true? Select one: a. R must be reflexive and transitive but might not be symmetric. b. R must be reflexive and symmetric but might not be transitive. C. R must be symmetric and transitive but might not be reflexive. d. R must be an equivalence relation, but { [x]R:XEX} might not be equal to P. e. R must be an equivalence relation, and {[x]R: XEX} must equal P.
Transcribed Image Text:QUESTION 8 Let X be a set. Let P be a set of subsets of X such that: • Ø & P; the union of all sets A E P is X. Note that these are clauses (a) 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 E P}, as in Theorem 1.7(b). Which of the following is true? Select one: a. R must be reflexive and transitive but might not be symmetric. b. R must be reflexive and symmetric but might not be transitive. C. R must be symmetric and transitive but might not be reflexive. d. R must be an equivalence relation, but { [x]R:XEX} might not be equal to P. e. R must be an equivalence relation, and {[x]R: XEX} must equal P.
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