Let \(A_1, A_2, \ldots , A_n\) be sets such that \(X = \bigcup_{i=1}^n A_i \). Prove that there exists a sequence of sets \(B_1, B_2, \ldots , B_n\) such that

a) \(B_i \subseteq A_i \) for each \(i=1,2,\ldots ,n\).

b) \(B_i \cap B_j = \varnothing \) for \(i\ne j\).

c) \(X = \bigcup_{i=1}^n B_i \).

Can you give insights on how to solve this problem? Insights is enough for me.

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## Comments

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TopNewestJust look at the "new" things added to \(X\) by \(A_i\). Call that \(B_i\). For example \(B_1\) is \(A_1\), \(B_2\) is \(A_2 \setminus A_1\), etc. Can you continue from here?

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Nope. I cannot still comprehend.

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What to do next.

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Now just verify that the claims given hold for these \(B_i\)'s

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That for (a) I will assume first that \( x\in A_k \) and do all ways to prove that \( x\in B_k \) given \(B_i \subseteq A_i \) for each \(i=1,2,\ldots ,n\)?

For (b), I need to prove that \( B_i \) is a disjoint set such that \(A_{i} \setminus B_{i-1} \bigcap A_{j} \setminus B_{j-1}=\emptyset \) ?

Lastly, for (c), I need to prove that \(B_i=B_k\) which implies that \(A_{i} \setminus B_{i-1} = A_{j} \setminus B_{j-1}\)?

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