- #1
Klungo
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Homework Statement
(Title is wrong)
I was able to prove the similar [itex] \bigcup(F \cup G)=(\cup F)\bigcup(\cup G) [/itex] but I'm not to sure how to go about this one.
Let F and G be nonempty families of sets. Prove
[itex] \bigcap(F \cup G)=(\cap F)\bigcap(\cap G) [/itex]
2. The attempt at a solution
To prove [itex]\bigcap(F \cup G) \subset (\cap F)\bigcap(\cap G)[/itex],
let [itex] x \subset \bigcap(F \cup G) [/itex] be arbitrary. Clearly, [itex] x \in S[/itex] for some set [itex] S \subset \bigcap F \cup G [/itex] containing all common elements in [itex] F \cup G [/itex]. We have [itex] S \in F \cup G.[/itex]
Do I go by..
We have two cases: (only one needs to be proven really.)
Case 1: [itex] If S \in F [/itex], clearly [itex] S \subset \cup F [/itex]...
(I'm stuck here)
or Do I go by...
Suppose [itex] S \in F [/itex] and [itex] S \in G [/itex]. Clearly [itex] S \subset \cup F [/itex] and [itex] S \subset \cup G [/itex]. (I'm stuck here)
I'll try proving the converse and restart the proof since I misread the problem.
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