Changes

The concept of a sign relation is typically extended as a set <math>\mathcal{L} \subseteq \mathcal{O} \times \mathcal{S} \times \mathcal{I}.</math>  Because this extensional representation of a sign relation is one of the most natural forms that it can take up, along with being one of the most important forms that it is likely to be encountered in, a good amount of set-theoretic machinery is necessary to carry out a reasonably detailed analysis of sign relations in general.

The concept of a sign relation is typically extended as a set <math>\mathcal{L} \subseteq \mathcal{O} \times \mathcal{S} \times \mathcal{I}.</math>  Because this extensional representation of a sign relation is one of the most natural forms that it can take up, along with being one of the most important forms that it is likely to be encountered in, a good amount of set-theoretic machinery is necessary to carry out a reasonably detailed analysis of sign relations in general.

For the purposes of this discussion, let it be supposed that each set <math>X,\!</math> that comprises a subject of interest in a particular discussion or that constitutes a topic of interest in a particular moment of discussion, is a subset of a set <math>U,\!</math> one that is sufficiently universal relative to that discussion or big enough to cover everything that is being talked about at that moment.  In a setting like this it is possible to make a number of useful definitions, to which I now turn.

For the purposes of this discussion, let it be supposed that each set <math>Q,\!</math> that comprises a subject of interest in a particular discussion or that constitutes a topic of interest in a particular moment of discussion, is a subset of a set <math>X,\!</math> one that is sufficiently universal relative to that discussion or big enough to cover everything that is being talked about in that moment.  In a setting like this it is possible to make a number of useful definitions, to which we now turn.

The ''negation'' of a sentence <math>S,\!</math> written as <math>^{\backprime\backprime} \, \underline{(} S \underline{)} \, ^{\prime\prime}</math> and read as <math>^{\backprime\backprime} \, \operatorname{not}\ S \, ^{\prime\prime},</math> is a sentence that is true when <math>S\!</math> is false and false when <math>S\!</math> is true.

The ''negation'' of a sentence <math>s,\!</math> written as <math>^{\backprime\backprime} \, \underline{(} s \underline{)} \, ^{\prime\prime}</math> and read as <math>^{\backprime\backprime} \, \operatorname{not}\ s \, ^{\prime\prime},</math> is a sentence that is true when <math>s\!</math> is false and false when <math>s\!</math> is true.

The ''complement'' of a set <math>X\!</math> with respect to the universe <math>U,\!</math> written as <math>^{\backprime\backprime} \, U\!-\!X \, ^{\prime\prime},</math> or simply as <math>^{\backprime\backprime} \, {}^{_\sim}\!X \, ^{\prime\prime}</math> when the universe <math>U\!</math> is understood, is the set of elements in <math>U\!</math> that are not in <math>X,\!</math> that is:

The ''complement'' of a set <math>Q\!</math> with respect to the universe <math>X\!</math> is denoted by <math>^{\backprime\backprime} \, X\!-\!Q \, ^{\prime\prime},</math> or simply by <math>^{\backprime\backprime} \, {}^{_\sim}\!Q \, ^{\prime\prime}</math> when the universe <math>X\!</math> is determinate, and is defined as the set of elements in <math>X\!</math> that do not belong to <math>Q,\!</math> that is:

{| align="center" cellpadding="8" width="90%"

{| align="center" cellpadding="8" width="90%"

|

|

<math>\begin{array}{lllll}

<math>\begin{array}{lllll}

{}^{_\sim}\!X

{}^{_\sim}\!Q

& = &

& = &

U\!-\!X

X\!-\!Q

& = &

& = &

\{ \, u \in U : \underline{(} u \in X \underline{)} \, \}.

\{ \, x \in X : \underline{(} x \in Q \underline{)} \, \}.

\\

\\

\end{array}</math>

\end{array}</math>