# Let $M$ be a non-empty set whose elements are sets. What are $F={A×{A} : A⊆M, A≠∅}$ and $⋃F$?

Mathematics Asked by Andrea Burgio on November 28, 2020

I think it’s not so difficult but I struggling a little to figure it out, I want to make sure I’m correct, is $$F$$ a set of the form:
$$F={ {(a, A), (b, A), …}, {(α, B), (β, B), …}, …}$$ for all $$a,b,…∈A$$ and $$α,β,…∈B$$, where $$A,B,…⊆M$$?

and $$⋃F$$ a set of the form:
$$⋃F={(a, A), (b, A), …,(α, B), (β, B), …}$$?

Thank you!

$$y$$ is an element of $$F$$ iff it has the shape $$y=Atimes{A}$$ where $$Asubseteq M$$ and $$Aneqvarnothing$$.

The equality in your title already states this and cannot essentially be improved, so you do not have to bother about $$F$$ itself.

$$x$$ is an element of $$bigcup F$$ iff $$xin y$$ for some $$yin F$$.

Referring to the first line we get:

$$x$$ is an element of $$bigcup F$$ iff $$xin Atimes{A}$$ for some $$A$$ with $$Asubseteq M$$ and $$Aneqvarnothing$$.

That means that $$x$$ must have the shape $$x=(a,A)$$ where $$ain A$$.

Proved is now that:$$bigcup F={(a,A)mid ain Asubseteq M}$$ The condition $$Aneqvarnothing$$ does not need to be mentioned here because $$ain A$$ implies that $$Aneqvarnothing$$.

Correct answer by drhab on November 28, 2020