Let me also try to point out the ambiguity in your question. When you say "consider ANY four subsets of the universal set" since the null set is a subset of every set then any four subsets of the universal set can mean (1) the null set and three arbitrary subsets (A,B,C) or (2) four arbitrary subsets sets (A,B,C,D) as shown above. In case (1) since the null subset has no region then the maximum number of regions that can be formed is 8 (you can use a Venn diagram as above to prove this). However, in case (2) you can see from the diagram above the maximum number of regions that can be formed is 14.
Furthermore, the question does not state maximum or minimum number of regions that can be formed. Take case (2) as an example there are 14 regions only because all the subsets intersect. If all four subsets where discrete then there would only be 5 regions and the same four subsets.
This kind of ambiguity is what discrete math is supposed to avoid. The questions and answers are to be so clear as to leave no room for interpretation!
Here is an example ....
An engineer, a physicist, and a mathematician were on a train heading north, and had just crossed the border into Scotland and pass by a field with one black sheep.
The engineer looked out of the window and said "Look! All Scottish sheep are black!"
The physicist said, "No, no, you are wrong. You can only say that some Scottish sheep are black."
The mathematician looking irritated says. "You are both wrong! All that can deduced is - In Scotland, there is at least one field, containing at least one sheep, of which at least one side is black."
"If I have seen further it is by standing on the shoulders of giants." Sir Isaac Newton, 1675
Last edited by ctbram : 11-02-2011 at 08:37 AM.