It seems intuitive that the power set of a union of sets P(XunionY) is not a subset of the union of the two respective power sets P(X)unionP(Y). For finite sets the former will have more elements than the latter.(adsbygoogle = window.adsbygoogle || []).push({});

However, I can't figure out what is wrong with the following line of reasoning:

For a set S

( S in P(XunionY) ) implies ( S is a subset of XunionY ) implies (1)

for some x

( x in S implies x in XunionY ) implies (2)

( x not in S OR x in X OR x in Y ) implies (3)

( ( x in S implies x in X ) OR (x in S implies x in Y) ) implies (4)

( ( S is a subset of X ) or ( S is a subset of Y ) ) (5)

I could continue, but I'm nearly certain I have made a mistake somewhere already. I can't figure out where. I suspect that (3) is incorrect, but I don't see why I can't make the substitution I made namely:

(A implies B) iff (notA OR B)

Also, I know that (5) is incorrect. Perhaps (5) doesn't follow from (4).

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# The power set of a union

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