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Ordered Sets

A set is ordered if its members obey three simple rules. First, an ordering relation such as "less than" (<) must apply to every member of the set, that is, for any two members of the set, call them a and b, either a < b or b < a. Second, no member of the set can have more than one position within the ordering, in other words, a < a has no meaning. Third, the ordering must be transitive, that is, for any three members of the set, call them a, b, and c, if a < b, and b < c, then a < c. There are many examples of ordered sets. The alphabet, for instance, is an ordered set whose members are letters. An encyclopedia is an ordered set whose members are entries that are ordered alphabetically. The real numbers and subsets of the real numbers are also ordered. As a consequence, any set that is ordered can be associated on a one-to-one basis with the real numbers, or one of its subsets. The algebra of inequalities, then, is applicable to any set regardless of whether its members are numbers, letters, people, dogs or whatever, as long as the set is ordered.

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Science EncyclopediaScience & Philosophy: Incomplete dominance to IntuitionismInequality - Ordered Sets, Algebra Of Inequalities, Examples