(b) Observe that all the relations in (a) are equivalence relations. Prove that this is true in general, that is, prove that if A is a nonempty set, and P is a partition of A, then the relation R corresponding to P defined in (*) must be an equivalence relation.
(b) Observe that all the relations in (a) are equivalence relations. Prove that this is true in general, that is, prove that if A is a nonempty set, and P is a partition of A, then the relation R corresponding to P defined in (*) must be an equivalence relation.
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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I just need answer for ii
Transcribed Image Text:*18. Let A be a nonempty set and let P be a partition of A. Define a relation R (corresponding
to P) on A by
X.
x Ry if there exists S E P such that x, y E S.
(*)
(a) Let A = {1, 2,3}. Write down ALL possible partitions of A. For each of the partition
P, use (*) to write down the relation R (as a subset of A x A) corresponding to P.
(b) Observe that all the relations in (a) are equivalence relations. Prove that this is true
in general, that is, prove that if A is a nonempty set, and P is a partition of A, then
the relation R corresponding to P defined in (*) must be an equivalence relation.
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