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Thread: Need Explanation through use of venn diagram

  1. #1 Need Explanation through use of venn diagram 
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    Hi, I'm really in a pinch. Can anyone help me understand this?
    Assume sets A and B are non-empty and unequal

    a)B-A=Null
    b) A U B = B
    c) A-B = A

    I hope someone can help me through venn diagrams, easier to understand. I searched for hours in the internet..either I suck at searching or there's just nothing...

    I appreciate any smart person who'd be able to help me with this..
    thanks..


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  3. #2  
    Your Mama! GiantEvil's Avatar
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    For B-A=Null, to be true requires that either, B be empty, or that B be a subset of A, or B=A.
    For A U B=B, to be true requires that either, A be a subset of B, or A=B, or A to be empty.
    For A-B=A, to be true requires that either, B be empty, or A be empty, or A and B to be disjoint.


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  4. #3  
    Forum Professor Daecon's Avatar
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    Quick question, what does "Null" and "U" mean in this context?

    It sounds like a logic puzzle but I'm unfamiliar with some of the notation.

    [edit]Nevermind, Wikipedia to the rescue.

    They appear to contradict each other, so they can't all be true at the same time.
    Last edited by Daecon; January 19th, 2014 at 04:59 PM.
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  5. #4  
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    Quote Originally Posted by Daecon View Post
    Quick question, what does "Null" and "U" mean in this context?

    It sounds like a logic puzzle but I'm unfamiliar with some of the notation.

    [edit]Nevermind, Wikipedia to the rescue.

    They appear to contradict each other, so they can't all be true at the same time.
    If A and B are empty, then all are true for the same A and B. I believe the three questions are actually independent of each other.
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  6. #5  
    Forum Professor Daecon's Avatar
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    Hmm, then it probably would have been better to label the sets as A & B, C & D, and E & F to avoid confusion.
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  7. #6  
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    I assume that are generic labels i.e. in the 3 exercises they may refer to different pairs of non-empty sets

    I also use standard set-theoretic notation - if you don't know it, look it up;

    1. (equality is excluded in the exercise) since if you remove the superset from its subset you are left with nothing

    2. (equality is again excluded) since if you join a subset to its superset you are left with the superset

    3. where is the complement of since any set and its complement are disjoint
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  8. #7  
    Forum Radioactive Isotope MagiMaster's Avatar
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    Minor nitpick, but I think that last one should be .
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  9. #8  
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    hi guys! thanks, I the first two are clear. as for the last one, uhm..three | Flickr - Photo Sharing!is this the correct way of illustrating it?
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  10. #9  
    Forum Radioactive Isotope MagiMaster's Avatar
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    That looks backwards, if I've understood your labeling correctly.
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  11. #10  
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    (i) A subset B

    (ii) A' = Null
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  12. #11  
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    Quote Originally Posted by MagiMaster View Post
    Minor nitpick, but I think that last one should be .
    Um....not too sure about this - lemme go on a ramble, though I am unsure of my logic.......

    I suppose a set and further suppose that its complement is any set that does NOT contain . Write the complement as and suppose this last statement implies that .

    I see 2 problems - first that this is a bit of a hollow definition. In that, any object whatever in the universe of objects that is definitely not, say, a red apple, is in - so all tractors, all trees, all real numbers, all functions, all stars etc etc are included in this set. Not very useful

    Worse, it leads rather directly to the Russell Paradox - colloquially one says this set is "too large to be a set"

    So let's refine the definition - we still insist that but now further insist there exist a set, say such that . One then says that is the complement of in .

    Now suppose that . Then I may insist that there exist the set say such that , so that is the complements of in . And so the game goes on and we end up with that is there exist just one that is definitely NOT a red apple (to use my earlier example). Again not very helpful

    That is why I wrote for the exercise that
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  13. #12  
    Forum Radioactive Isotope MagiMaster's Avatar
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    I've always assumed is taken as the complement with respect to some universe of discussion. So if your universe of discussion is the integers, would be all the integers not in (and it would make sense for ). But I hadn't thought about the possibility that the universe of discussion might not be a proper set (such as if your universe was all sets).
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  14. #13  
    Forum Professor river_rat's Avatar
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    Quote Originally Posted by MagiMaster View Post
    I've always assumed is taken as the complement with respect to some universe of discussion. So if your universe of discussion is the integers, would be all the integers not in (and it would make sense for ). But I hadn't thought about the possibility that the universe of discussion might not be a proper set (such as if your universe was all sets).
    That is the normal definition for standard set theory. Here is an easy example. Let A = Set of bolts, B = Set of canaries and C = Set of Birds union Set of fasteners. So both A and B are subsets of C (which is our universe for this discussion). Also A \ B = A as no canary is also a bolt. However, it is not true that B = C \ A (or A compliment) as C \ A also contains ostriches and screws, neither being canaries.
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  15. #14  
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    Maybe the question is written wrong but B is and empty box inside a box and A is just an empty box.

    That's my two cents anyway.

    Cardinality is one element.

    Cardinality is zero elements.

    :EDIT:

    No, it would screw things up... Those objects wouldn't be able to be combined then.
    Last edited by Beer w/Straw; January 27th, 2014 at 08:19 AM.
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  16. #15  
    Bassaricyon neblina Olinguito's Avatar
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    This way of stating the answer avoids complications with complements and universe of discussions.
    Last edited by Olinguito; January 30th, 2014 at 10:49 AM.
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  17. #16  
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    A is not empty and is in a box B (A subset B) - which is empty apart from having A (A'= Null) .
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