Proofs a logical form

Question

Let $\mathcal F$ be a non-empty family of sets with $A\in\mathcal F$.

$(a)$ Prove $A\subset\bigcup\mathcal F$

$(b)$ Prove $\cap\mathcal F\subset A$

$(c)$ Why was the assumption that $\mathcal F$ is nonempty needed?

Was it needed for both parts $(a)$ and $(b)$, or just one? If just one, which one?

Welcome to Mathematics Stack Exchange. I suppose if $A\in\mathscr F$, then it goes without saying that $\mathscr F$ is non-empty — J. W. Tanner, Mar 02 '20 at 04:49

score 0 · Answer 1 · answered Mar 02 '20 at 05:50

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a) Prove ⊂⋃ℱ

I did Let x ∈ A then x ∈ F since A ∈ F. Since x ∈ F, it follows that x ⊂ UF. Since x ∈ A then A ⊂ UF.

Is it correct?

answered Mar 02 '20 at 05:50

Kuan N

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$x\in A$ and $A\in F$ means $x\in\cup F$, but it does not mean $x\in F$ – J. W. Tanner Mar 02 '20 at 06:00

1 Answers1