A question related to sets, their intersections and unions.

Question

Which of the following is correct?

• a) $A\cup P(A)=P(A)$

• b) $A\cap P(A)=A$

• c) $A- P(A)=A$

• d) $P(A)-\{A\}=P(A)$ Here, $P(A)$ denotes power set of a set $A$.

Answer 1

a) and b) boil down to the claim that every element of $A$ is also a subset of $A$. c) and d) boil down to the claim that no element of $A$ is a subset of $A$. Both claims are false. Play with $A=\{\emptyset,x\}$ with generic $x$.

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By an edit, d) was changed rom $P(A)-A=P(A)$ to $P(A)-\{A\}=P(A)$. This is still false as it is equivalent to saying that $A$ (the unique element of $\{A\}$) is not a subset of $A$ (which it is, of course).