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− | The relationship between <math>Cond\!</math> and <math>L\!</math> exemplifies the standard association that exists between any binary operation and its corresponding triadic relation. | + | The relationship between <math>\operatorname{Cond}</math> and <math>L\!</math> exemplifies the standard association that exists between any binary operation and its corresponding triadic relation. |
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− | The conditional sign "<math>\rightarrow\!</math>" denotes the same formal object as the function name "<math>Cond\mbox{ }\!</math>", the only difference being that the first is written infix while the second is written prefix. Thus we have the following equation: | + | The conditional sign "<math>\rightarrow\!</math>" denotes the same formal object as the function name "<math>\operatorname{Cond}\mbox{ }\!</math>", the only difference being that the first is written infix while the second is written prefix. Thus we have the following equation: |
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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 ''[[image (mathematics)|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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− | | <math>L_{..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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| The same object is achieved in the following way. Begin with the binary operation: | | The same object is achieved in the following way. Begin with the binary operation: |
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− | : <math>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>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>T\!</math>, notated as follows: |
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− | : <math>Cond^{-1}(T) \subseteq \mathbb{B} \times \mathbb{B}\,.\!</math> | + | : <math>\operatorname{Cond}^{-1}(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>Cond^{-1}(T) = \{ (p,\ q) \in \mathbb{B} \times \mathbb{B}\ :\ Cond (p,\ q) = T \}\,.\!</math> | + | : <math>\operatorname{Cond}^{-1}(T) = \{ (p,\ q) \in \mathbb{B} \times \mathbb{B}\ :\ \operatorname{Cond} (p,\ q) = 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>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: |
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− | : <math>(p \Rightarrow q) \iff (p,\ q) \in L_{..T} \iff (p,\ q) \in Cond^{-1}(T)\,.\!</math> | + | : <math>(p \Rightarrow q) \iff (p,\ q) \in L_{..T} \iff (p,\ q) \in \operatorname{Cond}^{-1}(T)\,.\!</math> |
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| This completes the derivation of the mathematical objects that are denoted by the signs "<math>\rightarrow\!</math>" and "<math>\Rightarrow\!</math>" in this discussion. It needs to be remembered, though, that not all writers observe this distinction in every context. Especially in mathematics, where the single arrow sign "<math>\rightarrow\!</math>" is reserved for function notation, it is common to see the double arrow sign "<math>\Rightarrow\!</math>" being used for both concepts. | | This completes the derivation of the mathematical objects that are denoted by the signs "<math>\rightarrow\!</math>" and "<math>\Rightarrow\!</math>" in this discussion. It needs to be remembered, though, that not all writers observe this distinction in every context. Especially in mathematics, where the single arrow sign "<math>\rightarrow\!</math>" is reserved for function notation, it is common to see the double arrow sign "<math>\Rightarrow\!</math>" being used for both concepts. |