Equivalence classes and relations.

Quote:

Prove that an equivalence relation can on a set S uniquely defines a partition of that set.

Suppose S can be separated into $\displaystyle A_1, A_2,.......,A_n$ partitions.

Hence $\displaystyle A_1 \cup A_2 \cup........\cup A_n=S$. Assume that reflexivity, symmetry and transitivity hold.

1). Reflexivity $\displaystyle \Rightarrow A_1 \cap A_i=\phi$ since an element from one subset cannot belong to another subset (property 2 of a partition- the intersections are empty).

This is where I have no idea where to go. Symmetry and transitivity I cannot connect to the definition of a partition.

I was wondering if this is the right idea to follow.