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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.

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:

: <math>Cond : \mathbb{B} \times \mathbb{B} \to \mathbb{B}\,.\!</math>

: <math>\operatorname{Cond} : \mathbb{B} \times \mathbb{B} \to \mathbb{B}\,.\!</math>

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:

: <math>Cond^{-1}(T) \subseteq \mathbb{B} \times \mathbb{B}\,.\!</math>

: <math>\operatorname{Cond}^{-1}(T) \subseteq \mathbb{B} \times \mathbb{B}\,.\!</math>

This object is defined as follows:

This object is defined as follows:

: <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>

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:

: <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>

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.