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} \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>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:

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

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

& = &

& = &

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

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

\end{array}</math>

\end{array}</math>

|}

|}

The ''relative complement'' of <math>X\!</math> in <math>Y,\!</math> for two sets <math>X, Y \subseteq U,</math> written as <math>^{\backprime\backprime} \, Y - X \, ^{\prime\prime},</math> is the set of elements in <math>Y\!</math> that are not in <math>X,\!</math> 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:

Y?X = {u ? U : u ? Y and (u ? X) }.

Y - X

& = &

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

The "intersection" of X and Y, for two sets X, Y ? U, is denoted by "X ? Y" and defined as the set of elements in U that belong to both of X and Y.

The ''intersection'' of <math>X\!</math> and <math>Y,\!</math> for two sets <math>X, Y \subseteq U,</math> is denoted by <math>^{\backprime\backprime} \, X \cap Y \, ^{\prime\prime},</math> and defined as the set of elements in <math>U\!</math> that belong to both of <math>X\!</math> and <math>Y.\!</math>

X ? Y = {u ? U : u ? X and u ? Y }.

X \cap Y

& = &

\{ \, u \in U : u \in X\ \operatorname{and}\ u \in Y \, \}.

The "union" of X and Y, for two sets X, Y ? U, is denoted by "X ? Y" and defined as the set of elements in U that belong to at least one of X or Y.

The "union" of X and Y, for two sets X, Y ? U, is denoted by "X ? Y" and defined as the set of elements in U that belong to at least one of X or Y.