Can we conclude that a = b if a and b are two sets with the same power set?

Let’s think about this for a minute. If you were to take all of the elements in set A, and then remove one from each, would these new sets be equal? For example, lets say you have set A: {1, 2} and {1}. Then when removing 1 from each set you get: {2}, which is not equal to {1}.

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So in this case it seems like they are not equal. What about if you were to take all of the elements in set B, and then remove one from each? Would these new sets be equal? For example, lets say you have set A: {A} and {B}. Then when removing one letter from each set, we get: {} which is also not equal.

That means that a = b cannot be concluded based on two sets having the same power set because they may or may not end up being identical. So how can this conclusion be drawn for certain instead of just with “maybes”? One thing to consider is counting. If it turns out that the number of items in both powerset’s are exactly the same than this would serve as evidence supporting equality.

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