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Set Theory and Predicate Logic
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Induced Order
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The comparison operator < is what allows us to put real numbers into order to sort them. Another way to say this is to say that the usual ordering of the real numbers is the ordering induced by the < operator. By this point, you should be suspicious every time I use the word the, and I used it in the previous sentence in the ordering. Not every comparison operator on a set of things induces a well-de ned ordering, or an ordering at all, but less-than for numbers does.
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A Trichotomous > That Doesn t Induce an Ordering
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In the game rock-paper-scissors, the rules say that rock beats scissors, paper beats rock, and scissors beat paper. The idea of beats is a comparison, so we could de ne the > operator on the set {rock, paper, scissors} to mean beats, according to the game s rules. It shouldn t take you long to realize that it s not possible to order rock, paper, and scissors from best to worst according to the > operator. In this case, then, the ordering induced by the > operator is not well-de ned.
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The < operator for real numbers induces what mathematicians call a total order. To induce a total order, a comparison operator not only has to be trichotomous but also has to be antisymmetric and transitive. We ll take a look at these properties later.
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Ordinal Numbers
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Earlier, I de ned cardinality for nite sets. In particular, I observed that cardinality was well-de ned. The question The set S contains how many elements asks for a well-de ned answer, which might be The set S contains 10 elements. Notice how this question about cardinality and the sentence that answered it follow the pattern illustrated in Table 2-6.
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TABLE 2-6
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A Question Answered by a Cardinal Number
Sentence
The set S contains how many elements The set S contains how many elements The set S contains Solve for n: |S| = n. elements The set S contains 47 elements.
Description
Question sentence Question-word identi ed Question-word replaced by a ll-in-the-blank Blank lled in to produce the answer sentence Mathematical version of the question
Given a set S, the question in this case ( The set S contains how many elements ) has a well-de ned right answer. That s because the cardinality function, which answers how many elements questions about sets, is a well-de ned function. Numbers that answer a how many question are called cardinal numbers in mathematics because they express the cardinality of a set. The nite cardinal numbers are exactly the nonnegative integers, by the way, although
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there are many different in nite cardinal numbers. In nite sets are not all in nite in the same way, one could say. Unfortunately, we won t have a chance to look into that fascinating corner of mathematics here. Table 2-7 offers the same analysis of a similar question and its answer.
TABLE 2-7
A Question Answered by an Ordinal Number
Sentence
The number x appears in the list L in what position The number x appears in the list L in what position The number x appears in the list L in position The number x appears in the list L in the 47th position. None (explanation to follow).
Description
Question sentence Question-word identi ed Question-word replaced by a ll-in-the-blank Blank lled in to produce the answer sentence Mathematical version of the question
In the answer I gave, the number 47 (or the word 47th) is an ordinal number. In mathematics, an ordinal number is a number that can represent a position in order (as opposed to a cardinality). In the nite realm, the ordinal numbers and the cardinal numbers are the same, but we still have a reason to look at them separately.
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