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Prove that the equivalence classes of an equivalence relation form a partition of the domain of...

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Advance Math

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:

  1. Prove that every partition block is nonempty and every element? is in some block.
  2. Prove that if [?]??[?]???, then ???. Conclude that the sets[?]? for ??? are partitions of ?.
  3. Prove that ????[?]?=[?]?.

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