Changes

We have just met with the fact that the differential of the '''''and''''' is the '''''or''''' of the differentials.

We have just met with the fact that the differential of the '''''and''''' is the '''''or''''' of the differentials.

{| align="center" cellpadding="6" width="90%"

{| align="center" cellspacing="10" style="text-align:center; width:90%"

| align="center" |

|

<math>p ~\operatorname{and}~ q \quad \xrightarrow{~\operatorname{Diff}~} \quad \operatorname{d}p ~\operatorname{or}~ \operatorname{d}q</math>

<math>\begin{matrix}

|-

p ~\operatorname{and}~ q

| align="center" |

& \quad &

\xrightarrow{\quad\operatorname{Diff}\quad}

& \quad &

\operatorname{d}p ~\operatorname{or}~ \operatorname{d}q

\end{matrix}</math>

|}

 

{| align="center" cellspacing="10"

| [[Image:Cactus Graph PQ Diff ((dP)(dQ)).jpg|500px]]

|                                                 |

|        p q        --Diff-->    ((dp) (dq))   |

|}

|}

We begin with a proposition or a boolean function <math>f(p, q) = pq.\!</math>

We begin with a proposition or a boolean function <math>f(p, q) = pq.\!</math>

{| align="center" cellpadding="10"

{| align="center" cellspacing="10"

| [[Image:Venn Diagram F = P And Q.jpg|500px]]

| [[Image:Venn Diagram F = P And Q.jpg|500px]]

|-

|-

A function like this has an abstract type and a concrete type.  The abstract type is what we invoke when we write things like <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}</math> or <math>f : \mathbb{B}^2 \to \mathbb{B}.</math>  The concrete type takes into account the qualitative dimensions or the "units" of the case, which can be explained as follows.

A function like this has an abstract type and a concrete type.  The abstract type is what we invoke when we write things like <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}</math> or <math>f : \mathbb{B}^2 \to \mathbb{B}.</math>  The concrete type takes into account the qualitative dimensions or the "units" of the case, which can be explained as follows.

{| align="center" cellpadding="6" width="90%"

{| align="center" cellspacing="10" width="90%"

| Let <math>P\!</math> be the set of values <math>\{ \texttt{(} p \texttt{)},~ p \} ~=~ \{ \operatorname{not}~ p,~ p \} ~\cong~ \mathbb{B}.</math>

| Let <math>P\!</math> be the set of values <math>\{ \texttt{(} p \texttt{)},~ p \} ~=~ \{ \operatorname{not}~ p,~ p \} ~\cong~ \mathbb{B}.</math>

|-

|-

The first couple of operators that we need to consider are logical analogues of the pair that play a founding role in the classical finite difference calculus, namely:

The first couple of operators that we need to consider are logical analogues of the pair that play a founding role in the classical finite difference calculus, namely:

{| align="center" cellpadding="6" width="90%"

{| align="center" cellspacing="10" width="90%"

| The ''difference operator'' <math>\Delta,\!</math> written here as <math>\operatorname{D}.</math>

| The ''difference operator'' <math>\Delta,\!</math> written here as <math>\operatorname{D}.</math>

|-

|-

In order to describe the universe in which these operators operate, it is necessary to enlarge the original universe of discourse.  Starting from the initial space <math>X = P \times Q,</math> its ''(first order) differential extension'' <math>\operatorname{E}X</math> is constructed according to the following specifications:

In order to describe the universe in which these operators operate, it is necessary to enlarge the original universe of discourse.  Starting from the initial space <math>X = P \times Q,</math> its ''(first order) differential extension'' <math>\operatorname{E}X</math> is constructed according to the following specifications:

{| align="center" cellpadding="6" width="90%"

{| align="center" cellspacing="10" width="90%"

|

|

<math>\begin{array}{rcc}

<math>\begin{array}{rcc}

where:

where:

{| align="center" cellpadding="6" width="90%"

{| align="center" cellspacing="10" width="90%"

|

|

<math>\begin{array}{rcc}

<math>\begin{array}{rcc}

Drawing a venn diagram for the differential extension <math>\operatorname{E}X = X \times \operatorname{d}X</math> requires four logical dimensions, <math>P, Q, \operatorname{d}P, \operatorname{d}Q,</math> but it is possible to project a suggestion of what the differential features <math>\operatorname{d}p</math> and <math>\operatorname{d}q</math> are about on the 2-dimensional base space <math>X = P \times Q</math> by drawing arrows that cross the boundaries of the basic circles in the venn diagram for <math>X\!,</math> reading an arrow as <math>\operatorname{d}p</math> if it crosses the boundary between <math>p\!</math> and <math>\texttt{(} p \texttt{)}</math> in either direction and reading an arrow as <math>\operatorname{d}q</math> if it crosses the boundary between <math>q\!</math> and <math>\texttt{(} q \texttt{)}</math> in either direction.

Drawing a venn diagram for the differential extension <math>\operatorname{E}X = X \times \operatorname{d}X</math> requires four logical dimensions, <math>P, Q, \operatorname{d}P, \operatorname{d}Q,</math> but it is possible to project a suggestion of what the differential features <math>\operatorname{d}p</math> and <math>\operatorname{d}q</math> are about on the 2-dimensional base space <math>X = P \times Q</math> by drawing arrows that cross the boundaries of the basic circles in the venn diagram for <math>X\!,</math> reading an arrow as <math>\operatorname{d}p</math> if it crosses the boundary between <math>p\!</math> and <math>\texttt{(} p \texttt{)}</math> in either direction and reading an arrow as <math>\operatorname{d}q</math> if it crosses the boundary between <math>q\!</math> and <math>\texttt{(} q \texttt{)}</math> in either direction.

{| align="center" cellpadding="10"

{| align="center" cellspacing="10"

| [[Image:Venn Diagram P Q dP dQ.jpg|500px]]

| [[Image:Venn Diagram P Q dP dQ.jpg|500px]]

|}

|}