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Set Theory and Predicate Logic
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If you thought three-valued logic was a signi cant departure from the world of True and False, fuzzy logic is a further departure. The premise of fuzzy logic is that absolute truths or falsehoods aren t all we care about or know. We may decide to include a fact in our model that we are relatively certain of, but not absolutely so. In fuzzy logic, the discrete values False and True are replaced by the continuum of numbers from zero to one. A zero is an absolute falsehood, a one is an absolute truth, and in between are the shades of gray. A system can operate according to a threshold. You might only want to consider facts that are 99.5 percent likely to be true. Someone else might be willing to deal with 90 percent likelihood. Creating a rigorous mathematical framework for fuzzy logic is a serious challenge.
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Relations
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Operators such as = and <, which compare two elements of the same kind and yield a truth value as a result, are called relations. A relation ~ on elements of a set U can be considered as the set {(u,v) U U : u~u} of pairs of elements that satisfy the relation. Alternatively, ~ can be considered as a predicate with two variables, each of which has U as its domain.
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The Re exive, Symmetric, and Transitive Properties
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The de nition of > in the earlier rock-scissors-paper example wasn t typical. Most directional or bidirectional comparison operators in mathematics, such as <, , and =, are transitive. Here s a precise de nition of the transitive property and some other useful properties a relation can have.
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Properties of Relations
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Let ~ be a relation on the universal set U. In other words, let u ~ v have a well-de ned truth value whenever u and v are elements of U. The relation ~ is said to be re exive, irre exive, symmetric, antisymmetric, or transitive according to the following de nitions:
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Re exive Irre exive
The relation ~ is re exive if x~x is true for every x in U. The relation ~ is irre exive if x~x is false for every x in U.
Symmetric The relation ~ is symmetric if x~y and y~x always have the same truth value, when x and y are elements of U. Antisymmetric The relation ~ is antisymmetric if x~y and y~x always have the opposite truth value, when x and y are elements of U. Transitive The relation ~ is transitive if whenever x~y and y~z are true, x~z is also true, when x, y, and z are elements of U.
Inside Microsoft SQL Server 2008: T-SQL Querying
Although the names might suggest otherwise, it s not the case that every relation is either re exive or irre exive (or either symmetric or antisymmetric). An example of a relation that is neither re exive nor irre exive is the relation is the reverse of on words. There are words w for which w is the reverse of w, such as radar, but there are also words for which w is not the reverse of w, like sonar.
Not All < Operators Were Created Equal
Just as it was important to know a set s universe U, it s important to know a relation s universe it s part of what de nes the relation. The symbol < can appear between numbers or strings in SQL, but the relation < between numbers is not the same thing as the relation < between strings. If you aren t careful, as the following T-SQL example shows, you can run into trouble or at least what looks like trouble:
DECLARE @x VARCHAR(10); DECLARE @y INT; DECLARE @z VARCHAR(10); SET @x = '1000'; SET @y = '2000'; SET @z = '+3000'; SELECT CASE WHEN @x < @y THEN 'TRUE' ELSE 'FALSE' END AS [x<y ], CASE WHEN @y < @z THEN 'TRUE' ELSE 'FALSE' END AS [y<z ], CASE WHEN @x < @z THEN 'TRUE' ELSE 'FALSE' END AS [x<z ]
This produces the following output, which appears to contradict the transitivity of the T-SQL operator <.