Changes

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>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.

The ''negation'' of a sentence <math>S,\!</math> written as <math>^{\backprime\backprime} (S) ^{\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 X with respect to the universe U, written as "U?X", or?simply as "~X" when the universe U is understood, is the set of elements in U that are not in X, that is:

~X = U?X = {u ? U : (u ? X) }.

{}^{_\sim}\!X

& = &

U - X

& = &

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

The "relative complement" of X in Y, for two sets X, Y ? U, written as "Y?X", is the set of elements in Y that are not in X, that is:

The "relative complement" of X in Y, for two sets X, Y ? U, written as "Y?X", is the set of elements in Y that are not in X, that is: