My book on Sets says:
If A is a set, then P(A) = { X: X ⊆ A} is called the power set. It is the set of all subsets of A.
However the definition seems to say, every in P(A) is a subset of A, but they may be the same subset. Basically, the second sentence of the definition does not ring true to me.

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I'm not following what you problem is with this definition

It says, it is the set of all subsets of A. However, I don't see where 'all' comes from.

ie. all possibilities of subsets of a set

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