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

Let \(S\) be the equivalence relation arising out of the partition \(P\).

Concept  used of Equivalence relation

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

Show  the equivalence relation

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

Thus \(S\) is identical to the equivalence relation \(R\) that we started with.

The equivalence relation \(S\) arising from the partition \(P\) corresponding to a given equivalence relation \(R\) is the same as \(R\). ( \(S\) is identical to \(R\) ).