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 E P}, as in Theorem 1.7(b). Which of the following is true? Select one: a. R must be an equivalence relation, and { [x]R: XE X } must equal P. b. R must be reflexive and symmetric but might not be transitive. C. R must be reflexive and transitive but might not be symmetric. d. R must be symmetric and transitive but might not be reflexive. 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: • 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 E P}, as in Theorem 1.7(b). Which of the following is true? Select one: a. R must be an equivalence relation, and { [x]R: XE X } must equal P. b. R must be reflexive and symmetric but might not be transitive. C. R must be reflexive and transitive but might not be symmetric. d. R must be symmetric and transitive but might not be reflexive. 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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![Let X be a set. Let P be a set of subsets of X such that:
• ØP;
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) :xe A 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 an equivalence relation, and { [x]R: X EX} must equal P.
b. R must be reflexive and symmetric but might not be transitive.
c.
R must be reflexive and transitive but might not be symmetric.
d. R must be symmetric and transitive but might not be reflexive.
4
e.
R must be an equivalence relation, but { [x]R: X EX} might not be equal to P.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F69e63d15-e103-480b-854f-53f2f849e632%2F36919395-c257-4e91-888e-7c81ef8f22c7%2Fcenlwlj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let X be a set. Let P be a set of subsets of X such that:
• ØP;
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) :xe A 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 an equivalence relation, and { [x]R: X EX} must equal P.
b. R must be reflexive and symmetric but might not be transitive.
c.
R must be reflexive and transitive but might not be symmetric.
d. R must be symmetric and transitive but might not be reflexive.
4
e.
R must be an equivalence relation, but { [x]R: X EX} might not be equal to P.
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