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Can you conclude A = B if A and B are 2 sets with the same power set ?

Doubt : Is power set of a set unique ?

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P(A)=P(B) iff A=B

first understand what does it mean by equal set.

A={1,2,3}    B={2,1,3}   // They are equal sets, equal sets are nothing but same set is written twice(we know order of elements in set doesn't matter)

we can easily prove our claim

consider set are not equal but power set are

Let suppose A≠B means we have one element  X that ∈ A  but X∉ B so in that way {X} ∈ P(A) whereas {X} ∉ P(B)

But as we considered power set are equal so by contradiction we can say iff power set are equal set are equal.

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