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JonF
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If P(X) denotes the power set of X. Is |P(A)| = |P(B)| iff A=B true? If so, I have no idea how to prove the |P(A)| => |P(B)| iff A=B direction, so any hints would be great.
CRGreathouse said:Does |P(A)| = |P(B)| imply |A| = |B| for infinite sets without the CH?
Why do you know that? That's false for most bijections [itex]\mathcal{P}(A) \rightarrow \mathcal{P}(B)[/itex].CrankFan said:the restriction of [tex] g [/tex] to the set [tex]A_s[/tex], is a bijection from [tex]A_s[/tex] to [tex]B_s[/tex]
Yes, if the power set (the set of all subsets) of set A is equal to the power set of set B, then set A and set B must be equal.
Yes, it is possible for two different sets to have the same power set. For example, the sets {1,2} and {3,4} have the same power set of {{}, {1}, {2}, {1,2}} even though they are not equal.
Yes, the power set of the empty set is equal to the set containing only the empty set. In other words, the power set of an empty set is {{}}.
No, the power set of a set will always have more elements than the set itself. This is because the power set contains all possible subsets of the original set, including the empty set and the set itself.
The cardinality (number of elements) of a set is always smaller than the cardinality of its power set. This is because the power set contains all possible subsets, including the empty set and the set itself, which adds at least two elements to the power set.