No, you cannot conclude that set A is equal to set B just because they have the same power set.
The power set of any set is the set of all possible subsets of that set. While it is true that two sets having the same power set implies that they have the same number of elements, it does not guarantee that the elements themselves are the same. For instance, consider the following examples:
- Let A = {1, 2} and B = {3, 4}. Both sets have the same power set:
Power(A) = { {}, {1}, {2}, {1, 2} } and Power(B) = { {}, {3}, {4}, {3, 4} }. However, A and B are clearly not equal. - On the other hand, if two sets do have the same elements (for example, A = {x, y} and B = {x, y}), then they will also have the same power set.
In conclusion, while the power set can indicate that two sets might have the same number of elements, it does not confirm their equivalence in terms of the specific elements contained within those sets.