For all natural numbers, n, prove that the power set of A={1,...,n} has 2^n elements of the original set.

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Mmmmk.

So let P(n) be the propositional statement "the power set of A={1,...,n} has 2^n elements of A". P(0) = "The power set of A={1} has 2^0 elements", which seems to be true so P(0) holds. Now assume that P(j) is true for all j<k.

What now? I'm stuck.