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    Partial order relation

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    Let S = {0,1} and consider the partial order relation R defined on S X S X S as follows: for all ordered triples (a, b, c) and (d, e, f) in S X S X S.
    ( a, b, c ) R ( d, e, f ) <-> a &#8804; d, b &#8804; e, c &#8804; f,
    where &#8804; denotes the usual "less than or equal to" relation for real numbers. Do the maximal, greatest, minimal and least elements exist? If so, which are they?

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    Solution Preview

    The maximal and the greatest elements are the same element (1,1,1).
    Since for any (a,b,c)in SXSXS, a=0 or a=1, so a<=1; b=0 or b=1, so b<=1; c=0 or c=1, so c<=1. By the definition of R, (a,b,c)R(1,1,1).
    So (1,1,1) ...

    Solution Summary

    This shows how to determine if the maximal, greatest, minimal and least elements exist for a given situation.