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Math Help - Question re: unions

  1. #1
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    Question re: unions

    According to a book I'm reading (A Set Theory Workbook by Iain T. Adamson), if a is a set and belongs to a class B, then the union of B is a subset of a (at least, according to the book, this is provable in NBG).

    My question is, how could this be?

    Let's say that a = {1, 2}, b = {2, 3}, and B = {a, b}. My understanding is that the union of B = {1, 3}; thus, the union of B cannot be a subset of a because the number "3" does not belong to a.

    Yet here's a quote from the book I'm reading: "Suppose a belongs to B. Let x be any element of a. Since x belongs to a and a belongs to B it follows that x belongs to the union of B. So a is a subset of the union of B (page 79).

    What am I missing here?
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  2. #2
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    Quote Originally Posted by chuckg1982 View Post
    According to a book I'm reading (A Set Theory Workbook by Iain T. Adamson), if a is a set and belongs to a class B, then the union of B is a subset of a (at least, according to the book, this is provable in NBG).
    I do not know that textbook. But surely that is a typo.

    Quote Originally Posted by chuckg1982 View Post
    Let's say that a = {1, 2}, b = {2, 3}, and B = {a, b}. My understanding is that the union of B = {1, 3}; thus, the union of B cannot be a subset of a because the number "3" does not belong to a.
    Donít you mean \bigcup B  = \left\{ {1,2,3} \right\}~?

    Quote Originally Posted by chuckg1982 View Post
    Yet here's a quote from the book I'm reading: "Suppose a belongs to B. Let x be any element of a. Since x belongs to a and a belongs to B it follows that x belongs to the union of B. So a is a subset of the union of B (page 79).
    Is this the same text book?
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  3. #3
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    Quote Originally Posted by Plato View Post
    Don’t you mean \bigcup B  = \left\{ {1,2,3} \right\}~?
    I thought \bigcup B only included the elements that belonged to some (but not all) members of B? Or is it the elements that belong to some or all members of B?


    Is this the same text book?
    Yes.
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  4. #4
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    If it is the same textbook, then clearly one of those is a typo.

    Quote Originally Posted by chuckg1982 View Post
    I thought \bigcup B only included the elements that belonged to some (but not all) members of B? Or is it the elements that belong to some or all members of B?
    If \mathcal{B} is a collection of sets then the statement that
    x\in\bigcup \mathcal{B} means that \left( {\exists A \in \mathcal{B}} \right)\left( {x \in A} \right).
    That is the inclusive or just as ordinary union is the inclusive or.
    Last edited by Plato; June 13th 2011 at 09:26 AM.
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  5. #5
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    Re: Question re: unions

    The union is the set that includes all that is common amongst the unioned sets. One way that helped me understand it was to think of it as dissolving the sets. In your example the union of B would be the union of the set

    \bigcup\{a, b\} = \bigcup\{\{1, 2\}, \{2, 3\}\} = \{1, 2, 3\}

    Notice how we keep the outer brackets but dissolve the brackets for 'a' and 'b'. Thus, we keep all that is common amongst 'a' and 'b'. However, {1, 2, 3} is not a subset of {1, 2}. It is the other way around.
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    Re: Question re: unions

    Thank you.
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  7. #7
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    Re: Question re: unions

    I should probably rephrase what I said initially, because an intersection can be stated in a similar (and thus confusing) form: i.e., the intersection is what is common to each set. Therefore, the intersection of S, for some S = {a, b, c, ...} is what is common to each set a, b, c, ... In my initial phrasing, I am saying what is in the union of S is what is common among a, b, c; to be in the union is to be in any of a, b, c, ... Therefore, to rephrase, I should say the union is the set that includes all that is among the unioned sets. I believe this way better reflects the intuitive approach I had about dissolving the sets in the union, so that you are left with all that is among those sets.
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    Re: Question re: unions

    Quote Originally Posted by bryangoodrich View Post
    I should probably rephrase what I said initially, because an intersection can be stated in a similar (and thus confusing) form: i.e., the intersection is what is common to each set. Therefore, the intersection of S, for some S = {a, b, c, ...} is what is common to each set a, b, c, ... In my initial phrasing, I am saying what is in the union of S is what is common among a, b, c; to be in the union is to be in any of a, b, c, ... Therefore, to rephrase, I should say the union is the set that includes all that is among the unioned sets. I believe this way better reflects the intuitive approach I had about dissolving the sets in the union, so that you are left with all that is among those sets.
    But why not just learn and understand the definition?
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  9. #9
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    Re: Question re: unions

    Quote Originally Posted by Plato View Post
    But why not just learn and understand the definition?
    I could raise the philosophical point that one can only know definitions, not understand them, for there is nothing in them to understand. Definitions portray ideas, and it is in the ideas of our terms that we come to an understanding. For instance, one can know the algebraic definition of a dot product without having any understanding of the idea it conveys geometrically, which anyone can argue is the root of its very inception. However, to make such a point is to travel into needless pedantry. The point of elucidating the idea of a union on sets was merely to convey an intuitive comprehension I obtained, and that others I know have found fruitful. The more lay approach to conveying such an idea is so that one can have familiarity with the idea beyond just symbolic representation and perceiving its instances. If that leads one to better grasp its use, is that not learning and understanding the definition?
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