Prove that the equivalence classes of an equivalence relationform a partition of the domain of the relation. Namely, suppose ?be an equivalence relation on a set ? and define the equivalenceclass of an element ??? to be
[?]?:={???|???}.
That is [?]?=?(?). Divide your proof into the following threepeices:
- Prove that every partition block is nonempty and every element? is in some block.
- Prove that if [?]??[?]???, then ???. Conclude that the sets[?]? for ??? are partitions of ?.
- Prove that ????[?]?=[?]?.
