Prove that all set partitions are induced by equivalence relations.

Question

Here is the question:

Prove that if $P$ is a partition of a set $S$ then there exists an equivalence relation $R$ on $S$ such that $P = S/R$.

I'm very stuck with nowhere to go. Any Ideas?

Answer 1

If $P$ is a partition of $S$ then $S$ is divided into disjoint sets say $$S_1, S_2, ...,S_k.$$

Define your equivalence relation $R$ on $S$ as the following:

$x$ is related to $y$ iff $x$ and $y$ are in the same $ S_i$ for some $i$, $1\le i\le k$

Note that the equivalency classes are exactly the partition sets $S_i$ for $1\le i\le k$

Thus $P=S/R$