MyWikiBiz, Author Your Legacy — Sunday May 05, 2024
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, 00:24, 13 May 2012
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− | Associated with the triadic relation <math>L\!</math> is a binary relation <math>L_{\underline{~} ~ \underline{~} ~ \operatorname{T}} \subseteq \mathbb{B} \times \mathbb{B}</math> that is called the ''fiber'' of <math>L\!</math> with <math>\operatorname{T}</math> in the third place. This object is defined as follows: | + | Associated with the triadic relation <math>L\!</math> is a binary relation <math>L_{\underline{~} \, \underline{~} \, \operatorname{T}} \subseteq \mathbb{B} \times \mathbb{B}</math> that is called the ''fiber'' of <math>L\!</math> with <math>\operatorname{T}</math> in the third place. This object is defined as follows: |
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| {| align="center" cellspacing="10" width="90%" | | {| align="center" cellspacing="10" width="90%" |
− | | <math>L_{\underline{~} ~ \underline{~} ~ \operatorname{T}} = \{ (p, q) \in \mathbb{B} \times \mathbb{B} : (p, q, \operatorname{T}) \in L \}.</math> | + | | <math>L_{\underline{~} \, \underline{~} \, \operatorname{T}} = \{ (p, q) \in \mathbb{B} \times \mathbb{B} : (p, q, \operatorname{T}) \in L \}.</math> |
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| : <math>\operatorname{Cond} : \mathbb{B} \times \mathbb{B} \to \mathbb{B}\,.\!</math> | | : <math>\operatorname{Cond} : \mathbb{B} \times \mathbb{B} \to \mathbb{B}\,.\!</math> |
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− | Form the binary relation that is called the ''fiber'' of <math>\operatorname{Cond}</math> at <math>T\!</math>, notated as follows: | + | Form the binary relation that is called the ''fiber'' of <math>\operatorname{Cond}</math> at <math>\operatorname{T}</math>, notated as follows: |
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− | : <math>\operatorname{Cond}^{-1}(T) \subseteq \mathbb{B} \times \mathbb{B}\,.\!</math> | + | : <math>\operatorname{Cond}^{-1}(\operatorname{T}) \subseteq \mathbb{B} \times \mathbb{B}.</math> |
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| This object is defined as follows: | | This object is defined as follows: |
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− | : <math>\operatorname{Cond}^{-1}(T) = \{ (p,\ q) \in \mathbb{B} \times \mathbb{B}\ :\ \operatorname{Cond} (p,\ q) = T \}\,.\!</math> | + | : <math>\operatorname{Cond}^{-1}(\operatorname{T}) = \{ (p, q) \in \mathbb{B} \times \mathbb{B} : \operatorname{Cond} (p, q) = \operatorname{T} \}.</math> |
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| The implication sign "<math>\Rightarrow\!</math>" denotes the same formal object as the relation names "<math>L_{..T}\mbox{ }\!</math>" and "<math>\operatorname{Cond}^{-1}(T)\mbox{ }\!</math>", the only differences being purely syntactic. Thus we have the following logical equivalence: | | The implication sign "<math>\Rightarrow\!</math>" denotes the same formal object as the relation names "<math>L_{..T}\mbox{ }\!</math>" and "<math>\operatorname{Cond}^{-1}(T)\mbox{ }\!</math>", the only differences being purely syntactic. Thus we have the following logical equivalence: |