Let X be a set. Let P be a set of subsets of X such that: • ØEP; • if A and B are distinct elements of P, then ANB=0. Note that these are clauses (a) and (b) 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 tru Select one: O a. R must be an equivalence relation, and {[x]: xex) must equal P. O b. R must be an equivalence relation, but { [x]: xe X} might not be equal to P. 0 с. R must be reflexive and transitive but might not be symmetric. O d. R must be reflexive and symmetric but might not be transitive. O e. R must be symmetric and transitive but might not be reflexive.
Let X be a set. Let P be a set of subsets of X such that: • ØEP; • if A and B are distinct elements of P, then ANB=0. Note that these are clauses (a) and (b) 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 tru Select one: O a. R must be an equivalence relation, and {[x]: xex) must equal P. O b. R must be an equivalence relation, but { [x]: xe X} might not be equal to P. 0 с. R must be reflexive and transitive but might not be symmetric. O d. R must be reflexive and symmetric but might not be transitive. O e. R must be symmetric and transitive but might not be reflexive.
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:
• Ø&P,
• if A and B are distinct elements of P, then ANB=0.
Note that these are clauses (a) and (b) of the definition of a partition (Definition 1.5).
Now define a relation R on the set X by R={(x, y): xe A andy € 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, and {[x]: xEX) must equal P.
O b. R must be an equivalence relation, but { [x]: xe X} might not be equal to P.
O c. R must be reflexive and transitive but might not be symmetric.
O d. R must be reflexive and symmetric but might not be transitive.
O e.
R must be symmetric and transitive but might not be reflexive.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F98a2975c-b075-4e7f-95a3-3742d529c071%2F173e57c2-9e0d-4a73-8432-6f8193703f94%2Fc0c0ps8_processed.png&w=3840&q=75)
Transcribed Image Text:Let X be a set. Let P be a set of subsets of X such that:
• Ø&P,
• if A and B are distinct elements of P, then ANB=0.
Note that these are clauses (a) and (b) of the definition of a partition (Definition 1.5).
Now define a relation R on the set X by R={(x, y): xe A andy € 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, and {[x]: xEX) must equal P.
O b. R must be an equivalence relation, but { [x]: xe X} might not be equal to P.
O c. R must be reflexive and transitive but might not be symmetric.
O d. R must be reflexive and symmetric but might not be transitive.
O e.
R must be symmetric and transitive but might not be reflexive.
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