Help me with these sets anons

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Sorry for the bad pic here is another one

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This is getting retarded

think of the cardinality of both for finite sets.
Suppse |A| =n, |B|=m.
Is 2^n + 2^m = 2^(m+n)? Obviously not. Thus
P(A+B) tends to be a lot larger.
Showing that P(A) and P(B) is a subset of P(A+B) is pretty trivial

So your first statement is false, just pick the set A U B, it is an Element of P(A+B) but not of P(A) +P(B)

Ok nevermind i treated the direct sum of Sets as simple Union. If you defined the direct sum of Sets as the union of all elements in both sets, then both statements are true