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Math Help - Abstract Axiomatic System Problem - Proving a number of undefined term

  1. #1
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    Abstract Axiomatic System Problem - Proving a number of undefined term

    Consider the following axiom set, in which x's, y's, and "on" are the undefined terms:

    Axiom 1. There exist exactly five x's.
    Axiom 2. Any two distinct x's have exactly one y on both of them.
    Axiom 3. Each y is on exactly two x's.

    How many y's are there in the system? Prove your result.

    Solution (so far):

    I think, there the number of y's is undefined because Axiom 2 and Axiom 3 contradict each other.
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  2. #2
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by sedemihcra View Post
    Consider the following axiom set, in which x's, y's, and "on" are the undefined terms:

    Axiom 1. There exist exactly five x's.
    Axiom 2. Any two distinct x's have exactly one y on both of them.
    Axiom 3. Each y is on exactly two x's.

    How many y's are there in the system? Prove your result.

    Solution (so far):

    I think, there the number of y's is undefined because Axiom 2 and Axiom 3 contradict each other.
    I see no contradiction. How many ways are there to pick two distinct x's?
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  3. #3
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    Here is a model of the axiom set. The x's are in red.
    Attached Thumbnails Attached Thumbnails Abstract Axiomatic System Problem - Proving a number of undefined term-untitled.gif  
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  4. #4
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    Quote Originally Posted by undefined View Post
    I see no contradiction. How many ways are there to pick two distinct x's?
    There would be 10 ways to pick two distinct x's.
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  5. #5
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by sedemihcra View Post
    There would be 10 ways to pick two distinct x's.
    And there are 10 edges in the graph provided by Plato.
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    This chapter in my geometry book is before the finite geometry chapter, but it makes a little more sense now. Thanks all.
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  7. #7
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    I should point out that precisely because Plato was able to produce a model that satisfies the axioms (you can easily see that the model works by seeing the edges as y's), the axioms are consistent.
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  8. #8
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    Alternatively we can find a bijection between {1,...,10} and the 2-subsets of {1,...,5} which is what prompted my first response.

    In other words, label the x's and call the set X = {1,2,3,4,5}

    List the 2-subsets lexicographically as usual

    y1 = {1,2}
    y2 = {1,3}
    ...
    y10 = {4,5}

    See that this satisfies the axioms and that it is unique up to labeling.

    Plato's model looks nicer though.
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