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Math Help - Sets and Relations Question

  1. #1
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    Sets and Relations Question

    Let A be a subset of Z (integers) squared and, for (a,b),(c,d) in A, define (a,b) <= (c,d) if and only if a <= c and d <= b.

    A) Show that (A, <=) is a partially ordered set.
    B) Is (A, <=) totally ordered? Explain

    I am pretty confused in tackling this question, help is appreciated
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  2. #2
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  3. #3
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    For lexigraphic ordering shouldn't it be that (a,b) <= (c,d) iff a<=c and b<= d? How does this change the result in proving a PO set? And how do I prove that this is a PO set?
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  4. #4
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    Well take each of the three properties, check each out with the change.
    reflexive, anti-symmetric, and transitive.
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  5. #5
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    So therefore,
    A is reflexive because a <= a
    A is antisymmetric because if a <= c, c > a
    A is transitive because a <= c and a <= d therefore a <= d

    Is this correct or am I not showing it properly?
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  6. #6
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    Quote Originally Posted by kanoro View Post
    So therefore,
    A is reflexive because a <= a
    A is antisymmetric because if a <= c, c > a
    A is transitive because a <= c and a <= d therefore a <= d
    Is this correct or am I not showing it properly?
    Those are not pairs. This question is about ordering on a set of pairs.

    Are these true?
    \begin{gathered}<br />
  \left( {x,y} \right) \prec \left( {x,y} \right) \hfill \\<br />
  \left( {x,y} \right) \prec \left( {w,z} \right) \wedge \left( {w,z} \right) \prec \left( {x,y} \right)\, \Rightarrow \,\left( {x,y} \right) = \left( {w,z} \right) \hfill \\<br />
  \left( {x,y} \right) \prec \left( {w,z} \right) \wedge \left( {w,z} \right) \prec \left( {s,t} \right)\, \Rightarrow \,\left( {x,y} \right) \prec \left( {s,t} \right) \hfill \\ <br />
\end{gathered}
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  7. #7
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    Okay so the top one (x,y) < (x,y) . I don't think this would be true? Maybe I am just confused on the notation. The way I am going about it is that these would seem to be equal
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  8. #8
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    Also, how do the "define (a,b) <= (c,d) iff (a<=c) and d<=b" come into play? Where do account for a being less than or equal to c and d being less than or equal to b? I am still unsure how to show P.O. Anymore help is appreciated.
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  9. #9
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    I donít think that you understand this relation.
    I will show you the transitive property.

    Suppose that  \left( {a,b} \right) \prec \left( {c,d} \right)~\&~ \left( {c,d} \right) \prec \left( {e,f} \right) .

    That means that a\le c~\&~d\le b and c\le e~\&~f\le d.

    That gives a\le c\le e~\&~f\le d\le b so  \left( {a,b} \right) \prec \left( {e,f} \right) .
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