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Math Help - More Homework Help

  1. #1
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    More Homework Help

    Let C be the set of cities in the US with populations greater than 200,000 and let L be the relation on C defined by xLy (i.e., (x,y) are members of L) if and only if the population of x is not greater than the population of y. According to one website, |C|=107. Assume, we are looking at the official 2000 census figures

    (a) Is the following statement true or false? explain :
    There exists an n such that for every x then xLn



    (b) Prove or disprove that L is a partial order. There is one sublte issue here; use common sense in dealing with it.
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  2. #2
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    Quote Originally Posted by minkyboodle View Post
    Let C be the set of cities in the US with populations greater than 200,000 and let L be the relation on C defined by xLy (i.e., (x,y) are members of L) if and only if the population of x is not greater than the population of y. According to one website, |C|=107. Assume, we are looking at the official 2000 census figures

    (a) Is the following statement true or false? explain :
    There exists an n such that for every x then xLn



    (b) Prove or disprove that L is a partial order. There is one sublte issue here; use common sense in dealing with it.
    (a) \exists n : xLn \ \forall x \in C means that there is one city with the largest population. Since C is a finite set, this must be true.

    (b) Reflexiveness: \forall x \in C, xLx since the population of any city is certainly not greater than its own population.

    Anti-symmetry: Assume that there exist x,y \in C : x \leq y, y \leq x. Then, the population of town x is not greater than the population of town y, but the population of town y is also not greater than the population of town x. If you may, this gives us that Pop(x) \leq Pop(y), Pop(y) \leq Pop(x) \Rightarrow Pop(x) = Pop(y), so the populations of x and y are equal, however this certainly doesn't guarantee us that they are the same city, seeing as two cities may have the same population. Hence, there is no anti-symmetry.

    Can you do transitivity?
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