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Math Help - Positive sets

  1. #1
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    Positive sets

    I missed a class and the prof said we covered positive sets.

    I quickly looked it up on wiki'

    Positive and negative sets - Wikipedia, the free encyclopedia

    SO basically is it as simple as being that a positive set is a set with all elements in the set are positive?
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  2. #2
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    Re: Positive sets

    This Wiki page is about functional analysis and measure theory. Judging by your previous posts and this forum section, you are studying discrete mathematics. So your professor probably meant something different by positive sets.

    As you can see, many terms have several meanings in math, so the reliable way to find what is meant is to look in the textbook or lecture notes or ask your professor.
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    Re: Positive sets

    Here is a question on the assignment that will hopefully put it in context.

    1. Given 4 sets A,B,C,D, express the following in terms of positive sets.

    | AcCc |
    | ABCcD |
    | ABcD |
    | (BcCD)c |
    | AcBcCcD |
    Will that help you distinguish exactly what kind of positive sets we are dealing with here.
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  4. #4
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    Re: Positive sets

    Is that lowercase 'c' supposed to be subset?

    (EDIT: Er, no, that doesn't make sense either. I have no idea what this problem is asking...hopefully someone else does.)
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  5. #5
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    Re: Positive sets

    I am still not sure what positive sets are. I think that Ac means A^c, i.e., the complement of A. I wish people would write superscripts and subscripts if not in LaTeX, then at least in plain text using ^ and _ symbols.

    I found almost the same question from 2010. It's probably a definition in one particular textbook.

    My guess is that a positive set is an intersection of some of the given sets (not their complements), or something like this.
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    Re: Positive sets

    Just for personal reference, is | XYZ | here intended to mean \bigcup{X, Y, Z}, I assume? I've never seen mid brackets used like that before, and I don't know what it means when two non-tuple sets appear adjacent to one another, either.
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  7. #7
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    Re: Positive sets

    |A| usually means the cardinality of A. Adjacent sets probably mean intersection, but this needs to be checked.
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    Re: Positive sets

    Quote Originally Posted by Annatala View Post
    Just for personal reference, is | XYZ | here intended to mean \bigcup{X, Y, Z}, I assume? I've never seen mid brackets used like that before, and I don't know what it means when two non-tuple sets appear adjacent to one another, either.
    If you follow the link in emakarov's reply #5, it is clear that |A| is the cardinality of set A.
    Also that Ac=A^c the complement of A
    However, none of the rest makes any sense.

    Early in the 20th century, in topology R L Moore and his students used A+B as A\cup B and AB as A\cap B and since whole space was M then M-A was a complement. But surely that is not it.

    I have a hard time believing the original poster has no notes or textbook to follow. The reference to inclusion/exclusion in the link make me think that all of this related to counting.
    It may be something like this: |AC^c|=|A|-|A\cap C|.
    Last edited by Plato; November 23rd 2011 at 12:35 PM.
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    Re: Positive sets

    Quote Originally Posted by emakarov View Post
    |A| usually means the cardinality of A.
    Right, but clearly that's not what it means here. :P I'm assuming "positive set" is not the same thing as "finite cardinal", whatever a positive set may be.

    It looks like the other questioner left off the mid bars, so maybe it's something @ehpoc added for clarity (in vain), or otherwise not important.

    Quote Originally Posted by emakarov View Post
    Adjacent sets probably mean intersection, but this needs to be checked.
    Is that a common notation?
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  10. #10
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    Re: Positive sets

    Quote Originally Posted by Plato View Post
    If you follow the link in emakarov's reply #5, it is clear that |A| is the cardinally of set A.
    Thanks. I was so busy looking at the notation I didn't read the description at the bottom of the post.

    Quote Originally Posted by Plato View Post
    Early in the 20th century, in topology R L Moore and his students used A+B as A\cup B and AB as A\cap B and since whole space was M then M-A was a complement. But surely that is not it.
    They seem to be mixing \cup, +, -, and adjacency in the same problem (2). Unless + or adjacency is supposed to be symmetric difference, I have no idea how they get four binary set operators. The book would certainly have to describe all this somewhere.
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  11. #11
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    Re: Positive sets

    I posted it funny. I don't know how to utilize the tools to use all the proper math notation.

    Either way....

    From other assignments I can be sure that the c means compliment.

    I can also be sure that adjacent sets means intersect. And yes it is the cardinality of these sets.
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  12. #12
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    Re: Positive sets

    Okay, are the positive sets you are thinking about ehpoc required to not be complements of other sets?
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    Re: Positive sets

    1. Given 4 sets A,B,C,D, express the following in terms of positive sets.

    | AcCc |
    | ABCcD |
    | ABcD |
    | (BcCD)c |
    | AcBcCcD |
    My guess is that, given A, B, C, D, positive sets are intersections of some of these four sets (not their complements).

    Suppose that the universe has n elements. Then |A^cB^c|=|(A\cup B)^c|=n-(|A|+|B|-|AB|). Since ABC^cD\cup ABCD=ABD and ABC^cD and ABCD are disjoint, we have |ABC^cD|+|ABCD|=|ABD|.

    The last example is a little longer. As above, |A^cB^cC^cD|+|A^cB^cC^cD^c|=|A^cB^cC^c|, so it is sufficient to find the last two cardinalities. |A^cB^cC^cD^c|=n-|A\cup B\cup C\cup D|, and |A\cup B\cup C\cup D| can be expressed using inclusion-exclusion principle. |A^cB^cC^c| can be found similarly. Perhaps there is a more elegant solution.
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  14. #14
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    Re: Positive sets

    Quote Originally Posted by ehpoc View Post
    I posted it funny. I don't know how to utilize the tools to use all the proper math notation. From other assignments I can be sure that the c means compliment. I can also be sure that adjacent sets means intersect. And yes it is the cardinality of these sets.
    First a lesson is posting LaTeX.
    [TEX]A^c[/TEX] gives A^c
    [TEX]|A\cup B|=|A|+|B|-|AB|[/TEX] gives |A\cup B|=|A|+|B|-|AB|.
    [TEX](A\cap B)[/TEX] gives (A\cap B).

    Second, lets assume that the universe is finite, |\mathcal{U}|<\infty.

    Now |A^c|=|\mathcal{U}|-|A| so:

    |A^cC^c|=|(A\cup C)^c|=|\mathcal{U}|-[|A|+|C|-|AC|].

    If this is a correct reading of the problem, you finish.
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  15. #15
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    Re: Positive sets

    I missed the class where he introduced positive sets. I started this thread so someone could confirm what exactly is meant by positive sets so I could educate myself on them.

    I literally know nothing on solving this particular problem. I posted it so someone could confirm what I am supposed to be learning to solve it :P
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