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| | Nice to satisfy you, my title is Araceli Oquendo but I don't like when people use my complete title. To keep birds is one of the issues he loves most. Some time in the past he chose to live in Idaho. My job is a production and distribution officer and I'm performing fairly great financially.<br><br>Visit my site [http://Sportshop-union.de/index.php?mod=users&action=view&id=13058 sportshop-union.de] |
| In [[mathematics]] and [[computer science]], a '''dependency relation''' is a [[binary relation]] that is finite, [[symmetric relation|symmetric]], and [[reflexive relation|reflexive]]; i.e. a finite [[tolerance relation]]. That is, it is a finite set of ordered pairs <math>D</math>, such that
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| * If <math>(a,b)\in D</math> then <math>(b,a) \in D</math> (symmetric)
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| * If <math>a</math> is an element of the set on which the relation is defined, then <math>(a,a) \in D</math> (reflexive)
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| In general, dependency relations are not [[transitive relation|transitive]]; thus, they generalize the notion of an [[equivalence relation]] by discarding transitivity.
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| Let <math>\Sigma</math> denote the [[alphabet (computer science)|alphabet]] of all the letters of <math>D</math>. Then the '''independency''' induced by <math>D</math> is the binary relation <math>I</math>
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| :<math>I = \Sigma \times \Sigma \setminus D</math>
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| That is, the independency is the set of all ordered pairs that are not in <math>D</math>. Clearly, the independency is symmetric and irreflexive.
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| The pairs <math>(\Sigma, D)</math> and <math>(\Sigma, I)</math>, or the triple <math>(\Sigma, D, I)</math> (with <math>I</math> induced by <math>D</math>) are sometimes called the '''concurrent alphabet''' or the '''reliance alphabet'''.
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| The pairs of letters in an independency relation induce an equivalence relation on the [[free monoid]] of all possible strings of finite length. The elements of the [[equivalence class]]es induced by the independency are called [[trace monoid|traces]], and are studied in [[trace theory]].
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| ==Examples==
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| [[file:Relación de dependencia.svg|200px|right]]
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| Consider the alphabet <math>\Sigma=\{a,b,c\}</math>. A possible dependency relation is
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| :<math>\begin{matrix} D
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| &=& \{a,b\}\times\{a,b\} \quad \cup \quad \{a,c\}\times\{a,c\} \\
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| &=& \{a,b\}^2 \cup \{a,c\}^2 \\
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| &=& \{ (a,b),(b,a),(a,c),(c,a),(a,a),(b,b),(c,c)\}
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| \end{matrix}</math>
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| The corresponding independency is
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| :<math>I_D=\{(b,c)\,,\,(c,b)\}</math>
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| Therefore, the letters <math>b,c</math> commute, or are independent of one another.
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| [[Category:Mathematical relations]]
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Nice to satisfy you, my title is Araceli Oquendo but I don't like when people use my complete title. To keep birds is one of the issues he loves most. Some time in the past he chose to live in Idaho. My job is a production and distribution officer and I'm performing fairly great financially.
Visit my site sportshop-union.de