I already proved that P(A) ⋃ P(B) ⊆ P(A⋃B). Now I have to create a condition, "such that equality holds," and prove it works.
2026-04-11 03:27:43.1775878063
Find a general condition based on comparing sets A and B such that P(A) ∪ P(B) = P(A ∪ B) and show the condition works
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Hint:
If indeed $A,B$ are sets with: $$\wp(A)\cup\wp(B)=\wp(A\cup B)$$ then: $$A\cup B\in\wp(A\cup B)=\wp(A)\cup\wp(B)$$ so that: $$A\cup B\in\wp(A)\text{ or }A\cup B\in\wp(B)$$