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## Homework Statement

Incorrect Theorem: Suppose F and G are familes of sets. If [itex]\bigcup[/itex]F and [itex]\bigcup[/itex]G are disjoint, then so are F and G

a) What's wrong with the following proof of the theorem?

Proof. Suppose [itex]\bigcup[/itex]F and [itex]\bigcup[/itex]G are disjoint. Suppose F and G are not disjoint. Then we can choose some set A such that A[itex]\in[/itex]F and A[itex]\in[/itex]G. Since A[itex]\in[/itex]F, A[itex]\subseteq[/itex]F, so every element of A is in [itex]\bigcup[/itex]F. Similarly, since A[itex]\in[/itex]G, every element of Ais in [itex]\bigcup[/itex]G. But then every element of A is in both [itex]\bigcup[/itex]G and [itex]\bigcup[/itex]F, and this is impossible since [itex]\bigcup[/itex]F and [itex]\bigcup[/itex]G are disjoint. Contraddiction.

b)Find a counterexample to the theorem.

## Homework Equations

## The Attempt at a Solution

I've found a counterexample. If F and G are the empty set. F and G are not disjoint but [itex]\bigcup[/itex]F and [itex]\bigcup[/itex]G are.

i can't find why the proof is wrong though.

Thank you.