Let F be a partition of A. Prove there exists unique equivalence relation R such that F=A|R?

Question

Let F be a partition of A. Prove there exists unique equivalence relation R such that F=A|R?

I don't even know how to start.

I know to be a equivalence relation R must be reflexive, symmetric and transitive. but how to i prove that the Family F, a partition of A is equal to A|B?

Answer 1

Hint:

Define for $\;x,y\in A\;$ the following relation:

$$x\sim y\iff \exists\;T\in F\;\;s.t. \;\;x\in T\;\;\text{and also}\;\;y\in T$$

Prove the above is an equivalence relation and the partition's sets are precisely the equivalence classes.