Changes

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%"

{}^{_\sim}\!X

{}^{_\sim}\!X

& = &

& = &

U - X

U\!-\!X

& = &

& = &

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

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

|}

|}

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 <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:

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

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

|

|

<math>\begin{array}{lll}

<math>\begin{array}{lll}

Y - X

Y\!-\!X

& = &

& = &

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

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

|}

|}

The ''symmetric difference'' of <math>X\!</math> and <math>Y,\!</math> for two sets <math>X, Y \subseteq U,</math> written <math>^{\backprime\backprime} \, X + Y \, ^{\prime\prime},</math> is the set of elements in <math>U\!</math> that belong to just one of <math>X\!</math> or <math>Y.\!</math>

The ''symmetric difference'' of <math>X\!</math> and <math>Y,\!</math> for two sets <math>X, Y \subseteq U,</math> written <math>^{\backprime\backprime} \, X ~\hat{+}~ Y \, ^{\prime\prime},</math> is the set of elements in <math>U\!</math> that belong to just one of <math>X\!</math> or <math>Y.\!</math>

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

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

|

|

<math>\begin{array}{lll}

<math>\begin{array}{lll}

X + Y

X ~\hat{+}~ Y

& = &

& = &

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

\{ \, u \in U : u \in X\!-\!Y ~\operatorname{or}~ u \in Y\!-\!X \, \}.

\\

\\

\end{array}</math>

\end{array}</math>

|}

|}

The foregoing definitions are the bare essentials that are needed to get the rest of this discussion going, but they have to be regarded as almost purely informal in character, at least, at this stage of the game.  In particular, these definitions all invoke the undefined notion of what a ''sentence'' is, they all rely on the reader's native intuition of what a ''set'' is, and they all derive their coherence and their meaning from the common understanding, but the equally casual use and unreflective acquaintance, that just about everybody has of the logical connectives ''not'', ''and'', ''or'', as these are expressed in natural language terms.

The foregoing "definitions" are the bare essentials that are needed to get the rest of this discussion going, but they have to be regarded as almost purely informal in character, at least, at this stage of the game.  In particular, these definitions all invoke the undefined notion of what a "sentence" is, they all rely on the reader's native intuition of what a "set" is, and they all derive their coherence and their meaning from the common understanding, but the equally casual use and unreflective acquaintance, that just about everybody has of the logical connectives "not", "and", "or", as these are expressed in natural language terms.

As formative definitions, these initial postulations neither acquire the privileged status of untouchable axioms and infallible intuitions nor do they deserve any special suspicion, at least, nothing over and above the reflective critique that one ought to apply to all important definitions.  These dim beginnings of anything approaching genuine definitions also serve to accustom the mind's eye to a particular style of observation, that of seeing informal concepts presented in a formal frame, in a way that almost demands their increasing clarification.  In this style of examination, the frame of the set-builder expression "{u ? U : ... }" functions like the "eye of the needle" that one is trying to get a suitably rich mathematics through.

As formative definitions, these initial postulations neither acquire the privileged status of untouchable axioms and infallible intuitions nor do they deserve any special suspicion, at least, nothing over and above the reflective critique that one ought to apply to all important definitions.  These dim beginnings of anything approaching genuine definitions also serve to accustom the mind's eye to a particular style of observation, that of seeing informal concepts presented in a formal frame, in a way that almost demands their increasing clarification.  In this style of examination, the frame of the set-builder expression <math>\{ u \in U : \underline{~~~} \}</math> functions like the ''eye of the needle'' that one is trying to get a suitably rich mathematics through.

Part the task of the remaining discussion is to gradually formalize the promissory notes that are represented by these terms and stipulations and to see whether their casual comprehension can be converted into an explicit subject matter, one that depends on grasping the corresponding collection of almost wholly, if still partially, formalized conceptions.  To this I now turn.

Part the task of the remaining discussion is to gradually formalize the promissory notes that are represented by these terms and stipulations and to see whether their casual comprehension can be converted into an explicit subject matter, one that depends on grasping the corresponding collection of almost wholly, if still partially, formalized conceptions.  To this I now turn.

The "binary domain" is the set B = {0, 1} of two algebraic values, whose arithmetic follows the rules of GF(2).

The "binary domain" is the set B = {0, 1} of two algebraic values, whose arithmetic follows the rules of GF(2).