Changes

Let <math>\mathbb{B} = \{ \operatorname{F}, \operatorname{T} \}</math> be the ''[[boolean domain]]'' of two logical values.  The truth table shows the ordered triples of a [[triadic relation]] <math>L \subseteq \mathbb{B} \times \mathbb{B} \times \mathbb{B}\!</math> that is defined as follows:

Let <math>\mathbb{B} = \{ \operatorname{F}, \operatorname{T} \}</math> be the ''[[boolean domain]]'' of two logical values.  The truth table shows the ordered triples of a [[triadic relation]] <math>L \subseteq \mathbb{B} \times \mathbb{B} \times \mathbb{B}\!</math> that is defined as follows:

: <math>L = \{(p,\ q,\ r) \in \mathbb{B} \times \mathbb{B} \times \mathbb{B}\ :\ Cond (p,\ q)\ = r \}\,.\!</math>

| <math>L = \{ (p, q, r) \in \mathbb{B} \times \mathbb{B} \times \mathbb{B} : \operatorname{Cond}(p, q) = r \}.</math>

Regarded as a set, this triadic relation is the same thing as the binary operation:

Regarded as a set, this triadic relation is the same thing as 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>

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>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>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:

: <math>(p \rightarrow q) = Cond (p,\ q)\,.\!</math>

| <math>(p \rightarrow q) = \operatorname{Cond}(p, q).</math>

Consider once again the triadic relation <math>L \subseteq \mathbb{B} \times \mathbb{B} \times \mathbb{B}\!</math> that is defined in the following equivalent fashion:

Consider once again the triadic relation <math>L \subseteq \mathbb{B} \times \mathbb{B} \times \mathbb{B}\!</math> that is defined in the following equivalent fashion:

: <math>L = \{(p,\ q,\ Cond (p,\ q))\ :\ (p,\ q) \in \mathbb{B} \times \mathbb{B} \}\,.\!</math>

| <math>L = \{ (p, q, \operatorname{Cond}(p, q) ) : (p, q) \in \mathbb{B} \times \mathbb{B} \}.</math>

Associated with the triadic relation <math>L\!</math> is a binary relation <math>L_{..T} \subseteq \mathbb{B} \times \mathbb{B}\!</math> that is called the ''[[image (mathematics)|fiber]]'' of <math>L\!</math> with <math>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 ''[[image (mathematics)|fiber]]'' of <math>L\!</math> with <math>\operatorname{T}</math> in the third place.  This object is defined as follows:

: <math>L_{..T} = \{ (p,\ q) \in \mathbb{B} \times \mathbb{B}\ :\ (p,\ q,\ T) \in L \}\,.\!</math>

| <math>L_{..T} = \{ (p, q) \in \mathbb{B} \times \mathbb{B} : (p, q, \operatorname{T}) \in L \}.</math>

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: