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How can I evaluate the expression $ (A_1 \cup \ldots \cup A_n) \cap (B_1\cup \ldots \cup B_m) \cap (C_1 \cup \ldots \cup C_o) ?$

I know that for $(A_1 \cup \ldots \cup A_n) \cap B_1$ we get $((A_1\cap B_1) \cup \ldots \cup (A_n\cap B_1))$, but don't knw how the generalize this to the above.

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    In what sense do you mean "evaluate" - and in what context?2012-12-29

1 Answers 1

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  1. Step 1: Replace $A=A_1\cup\ldots\cup A_n$, and similarly $B$ and $C$. We have $A\cap B\cap C$.
  2. Step 2: Simplify. In this case this is simplified as it is.
  3. Step 3: Put $A_1\cup\ldots\cup A_n$ back in, add parenthesis. We have $((A_1\cup\ldots\cup A_n)\cap B)\cap C$.
  4. Step 4: Distribute. We have $(A_1\cap B\cup\ldots\cup A_n\cap B)\cap C$.
  5. Step 5: Restore the $B_1\cup\ldots\cup B_m$. The indices are now long and tiresome.
  6. Step 6: Distribute $A_i\cap(B_1\cup\ldots\cup B_m)$.
  7. Step 7: Distribute the intersection with $C$. Arrive at the union small expressions of the form $(A_i\cap B_j\cap C)$.
  8. Step 8: Place the union of $C_k$'s into the expression.
  9. Step 9: Distribute.
  10. Step 10: Have a beer and recall how much bookkeeping after indices is terrible!

We end up with the union of $A_i\cap B_j\cap C_k$ for $i\leq n, j\leq m, k\leq o$.

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    @Mark: Yes, this is very true. I think that it's an excellent approach for the general case. I also think that everyone should at least one time keep track of indices, so they'll know how much it is preferred to be avoided! :-)2012-12-29