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mliuzzolino
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Homework Statement
Let [itex] {B_j: j \in J} [/itex] be an indexed family of sets. Show that [itex] \bigcup_{i \in J} B_i \subseteq \bigcap_{j \in J} B_j [/itex] iff for all i, j, [itex] \in [/itex] J, Bi = Bj.
Homework Equations
The Attempt at a Solution
First show that [itex] \bigcup_{i \in J} B_i \subseteq \bigcap_{j \in J} B_j \Rightarrow [/itex] for all i, j, [itex] \in [/itex] J, Bi = Bj.
By contrapositive, [itex] B_i \neq B_j \Rightarrow \bigcup_{i \in J} B_i \not\subset \bigcap_{j \in J} B_j [/itex]
Suppose [itex] B_i \neq B_j [/itex].
Let [itex] x \in \bigcup_{i \in J} B_i [/itex]. So there exists an i in J such that x in Bi. But since Bi [itex]\neq[/itex] Bj, [itex]i \neq j[/itex] and there exists an [itex] i \in J \ni x \notin B_j. [/itex].
I know that the definition of the index family of intersections is for all j in J, x in Ej. But I'm not sure how to say that this isn't the case in the above proof...
Any guidance for a lost soul?