Prove that or any partition P of S, there is an equivalence relation on S whose equivalence classes are the elements of P.

Question

Prove that or any partition P of S, there is an equivalence relation on S whose equivalence classes are the elements of P.

However its converse is easy,any hints how to begin the proof. I guess i have to define a equivalence relation wisely.

Answer 1

Hint

Suppose that the partition is $P= \{p_i \mid i \in I\}$.

Define $x \sim y$ if and only if $x,y$ are in the same $p_i$. Prove that this is an equivalence relation having the required properties.