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 A and yA for some A € P), as in Theorem 1.7(b). Which of the following is true? Select one: O a. R must be reflexive and transitive but might not be symmetric. O b. R must be reflexive and symmetric but might not be transitive. O c. R must be symmetric and transitive but might not be reflexive. O d. R must be an equivalence relation, and ( [xle:xx) must equal P. Qe R must be an equivalence relation, but ( [xle:xx) might not be equal to P.
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 A and yA for some A € P), as in Theorem 1.7(b). Which of the following is true? Select one: O a. R must be reflexive and transitive but might not be symmetric. O b. R must be reflexive and symmetric but might not be transitive. O c. R must be symmetric and transitive but might not be reflexive. O d. R must be an equivalence relation, and ( [xle:xx) must equal P. Qe R must be an equivalence relation, but ( [xle:xx) might not be equal to P.
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 € A and y € A for some A E P), as in Theorem 1.7(b). Which of the following is true?
Select one:
O a. R must be reflexive and transitive but might not be symmetric.
O b. R must be reflexive and symmetric but might not be transitive.
O c. R must be symmetric and transitive but might not be reflexive.
O d. R must be an equivalence relation, and {[x]: xe X) must equal P.
O e. R must be an equivalence relation, but { [x]: xX} might not be equal to P.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F98a2975c-b075-4e7f-95a3-3742d529c071%2F62ea446b-bbae-467d-b88d-f8b65ccf8823%2F5n4m0so_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:
• 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 € A and y € A for some A E P), as in Theorem 1.7(b). Which of the following is true?
Select one:
O a. R must be reflexive and transitive but might not be symmetric.
O b. R must be reflexive and symmetric but might not be transitive.
O c. R must be symmetric and transitive but might not be reflexive.
O d. R must be an equivalence relation, and {[x]: xe X) must equal P.
O e. R must be an equivalence relation, but { [x]: xX} might not be equal to P.
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