Let A be a set. A partition of A is a collection P = {Ai}i∈I of nonempty subsets of A which satisfies: a) Ai ∩ Aj = ∅ if i 6= j. b) A = S i∈I Ai In class, we proved that if ∼ is an equivalence relation on A, then A/ ∼ is a partition of A. Suppose P = {Ai}i∈I is a partition of A. Define a relation ∼P on A as follows: We say x ∼P y if and only if x and y are both members of the same set Aj ∈ P. Prove that A/ ∼P= P

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