Changes

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.

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} \, \texttt{(} s \texttt{)} \, ^{\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>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:

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} \, \tilde{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} Q

\tilde{Q}

& = &

& = &

X\!-\!Q

X\!-\!Q

& = &

& = &

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

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

\\

\\

\end{array}</math>

\end{array}</math>

Q\!-\!P

Q\!-\!P

& = &

& = &

\{ \, x \in X : x \in Q ~\operatorname{and}~ \underline{(} x \in P \underline{)} \, \}.

\{ \, x \in X : x \in Q ~\operatorname{and}~ \texttt{(} x \in P \texttt{)} \, \}.

\\

\\

\end{array}</math>

\end{array}</math>