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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4e66f5dc-fab6-4d54-ae8c-03cad0e4714b%2F9c02a149-01e6-4f4b-a553-7b3be7d9e8c9%2Fqed98by_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:
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.
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