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Old 10-02-2011, 08:56 PM   #1
silverspawn
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Default Maths Help

hey,
currently doing a discreet maths course and got on to the topic of set theory, I was wondering if someone could tell me weather I have the right answer to this questions I have answered.

Considering any 4 subsets of a universe, U, work out (and show your
working)
(1) how many different regions there could be.

My answer was 8 different regions.

(vi) how many different ways of combining these regions using boolean com-
binations are there?

I got this to 256 combining these regions.

I know this may be plain english to some people,But it has been so long since I touch on mathmatics

Thnaks in advance

Jason
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Old 11-02-2011, 05:36 AM   #2
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Well I would like to see the exact question because it sounds like what you posted has been paraphrased or translated poorly. If the question is - Given any four sets (A,B,C,D) within the universal region (U) what is the maximum number of possible regions? Then you would have 14 possible regions (see the image below).

As for the second part of the question, it does not make complete sense to me. A set is normally a well defined set of elements, the number of boolean results of which can be finitely defined. The sets in your question are not defined at all. Therefore the best I could do is define all the boolean operations you could perform on 4 sets. Such as:

1. compliment each (4)
2. AND pairs, then tuples, then 4's (intersection) ... (6), (3), (1)
3. OR pairs, then tuples, then 4's (union) ... (6) (3) (1)

Then you can AND and OR combinations of compliments and so forth. But as I said I am not 100% sure of the question as it seems ambiguous to me.
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Old 11-02-2011, 08:18 AM   #3
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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."
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Old 11-02-2011, 05:28 PM   #4
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And stwert looks even more irritated and says "no no... you're all wrong. All that can be INFERRED is that we have spotted what appears to be a sheep in what we believe is Scotland (based on certain geographical cues) and said organism (which might be in fact a sheep) has pigmentation that to our untrained eyes is darker than the other variety which humans label (but are in fact not at all, based on non-absolute proportions of reflected light-waves) black and white, respectively. All this is predicated on the assumption that we are in fact, conscious and also are not under the influence of any reality-distorting substances. That is all."

As for your questions... sorry, it's a bit over my head. I could try to give them a shot, but I'd rather write unhelpful remarks. All in good fun.
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Old 11-02-2011, 06:44 PM   #5
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Unfortunately that was all I was given in terms of the questions, which was given with some powerpoint slides as examples about set theory, which are pretty vague.

Jason
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Old 11-02-2011, 08:40 PM   #6
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No problem it is not uncommon to see questions like this. There may be some clues in the other lecture notes and text on the sections you are in that might infer more detail then the question as posted. Is your class being taught by the professor or a teaching assistant?

I hope you see my point regarding saying "any four subsets of the universal set" possibly containing the the null set and also there is nothing to specify whether the subsets intersect which would change the number of regions. So I am pretty sure I could prove any number of regions from 4 to 14 just depending on if I include the null set and how I chose to intersect the subsets.
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Old 17-02-2011, 10:42 AM   #7
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I emailed my lecturer and his response was "The clue to this is to represent the different possible subsets using binary "

sorry for the late reply I was without internet all week
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