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Math Help - Partial Orders and Cardinality

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    Partial Orders and Cardinality

    Just a question I was having some trouble with, any thoughts would be appreciated.


    - Suppose S is a (finite) set containing at least two elements for A,B that are elements of P(S), we define A is a poset on B to mean |A| =< |B|. Is the relation a partial order on P(S) and explain.


    Thanks!
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by kevi555 View Post
    Just a question I was having some trouble with, any thoughts would be appreciated.


    - Suppose S is a (finite) set containing at least two elements for A,B that are elements of P(S), we define A is a poset on B to mean |A| =< |B|. Is the relation a partial order on P(S) and explain.


    Thanks!
    to determine if we have a partial ordering, we must verify that the set has three properties: reflexivity, anti-symmetry and transitivity. that is, we must show that:

    1. A \le A for all A \in P(S)

    2. For all A,B \in P(S), If A \le B and B \le A then A = B

    3. For all A,B,C \in P(S), If A \le B and B \le C then A \le C


    and it seems that you have defined A \le B to mean |A| \le |B|

    by the way, what does "poset" mean?
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    poset meaning Partial ordered set...its symbol is the curly =< sign.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by kevi555 View Post
    poset meaning Partial ordered set...its symbol is the curly =< sign.
    i don't get what you are saying. i'm not familiar with the symbol you are talking about. you mean the less than or equal sign with a wavy line under it? in any case, i think that's just giving us a criterion by which to compare any two elements in the set in terms of the \le relation.

    i stick by my answer. check if the properties i've mentioned hold.
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    Quote Originally Posted by kevi555 View Post
    Suppose S is a (finite) set containing at least two elements for A,B that are elements of P(S), we define A is a poset on B to mean |A| =< |B|. Is the relation a partial order on P(S) and explain.
    It appears that you are confused about the terminology of ‘posets’.
    I think what the problem surely must be is the following
    Suppose S is a (finite) set containing at least two elements, define a relation on P(S) by this ARB if and only if |A|<|B|. Now the question is if (P(S), R) is a partial ordering or a poset?
    Or, is R a partial order on P(S)?
    Now clearly ordinary comparison of counting numbers is both reflexive and transitive.
    Your task is to decide if it is antisymmetric.
    HINT: If I were you, I would look for a simple counterexample.

    Post Script to Jhevon.
    |A| is the cardinality of the set A.
    ‘poset’ is a standard term, it means partially ordered set.
    Last edited by Plato; November 6th 2007 at 03:20 PM.
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