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 followingis 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.4e.R must be an equivalence relation, but { [x]R: X EX} might not be equal to P.

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Answered: Let X be a set. Let P be a set of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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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 E P is X. Note that these are clauses (a) and (c) of the definition of a partition Now define a relation R on the set X by R = {(x, y) :x EA and y E A for some A EP}, as in Theorem 1.7(b). Which of the following is true? Select one: O a. R must be symmetric and transitive but might not be reflexive. O b. R must be reflexive and symmetric but might not be transitive. C. R must be an equivalence relation, and { [x]R:X EX} must equal P. O d. R must be reflexive and transitive but might not be symmetric. e. R must be an equivalence relation, but { [x]R:X EX} might not be equal to P.
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