Question

To determine the partition \(Q\) arising from equivalence relation \(R\) corresponding to a given partition \(P\).

Short answer

\(Q\) and \(P\) define the same partitions

Step-by-step solution

Given data

Let \(P\) be any partition and \(R\) be the corresponding equivalence relation.

Let \(Q\) be the partition arising out of the equivalence relation \(R\).

Concept  used of Equivalence relation

An equivalence relation is a binary relation that is reflexive, symmetric and transitive.

Show  the equivalence relation

By definition, \(x\) and \(y\) are in the same subset of \(Q\) if and only if \(x\) is related to y through \(R\) which happens if and only if \(x\) and \(y\) are in the same subset defining the partition \(P\).

Thus \(Q\) is identical to partition \(P\) that we started with.

\(Q\) and \(P\) defines the same partitions: we obtain the same partition that we started with.