Changes

Intuitively, we want to map this <math>(\mathbb{B}^2)^n</math> onto <math>\mathbb{D}^n</math> by mapping each component <math>\mathbb{B}^2</math> onto a copy of <math>\mathbb{D}.</math>  But in the presenting context "<math>\mathbb{D}</math>" is just a name associated with, or an incidental quality attributed to, coefficient values in <math>\mathbb{B}</math> when they are attached to features in <math>\operatorname{d}\mathcal{X}.</math>

Intuitively, we want to map this <math>(\mathbb{B}^2)^n</math> onto <math>\mathbb{D}^n</math> by mapping each component <math>\mathbb{B}^2</math> onto a copy of <math>\mathbb{D}.</math>  But in the presenting context "<math>\mathbb{D}</math>" is just a name associated with, or an incidental quality attributed to, coefficient values in <math>\mathbb{B}</math> when they are attached to features in <math>\operatorname{d}\mathcal{X}.</math>

Taking these inetntions into account, define <math>\operatorname{d}x_i : X^2 \to \mathbb{B}</math> in the following manner:

Taking these intentions into account, define <math>\operatorname{d}x_i : X^2 \to \mathbb{B}</math> in the following manner:

:{| cellpadding=2

: <p><math>\begin{array}{lcrcl}

| d''x''_''i''(‹''u'', ''v''›)

\operatorname{d}x_i ((u, v)) & = & (\!|\ x_i (u) & , & x_i (v)\ |\!) \\

| =

                            & = & x_i (u)       & + & x_i (v)       \\

| <font face=system>(</font> ''x''_''i''(''u'') , ''x''_''i''(''v'') <font face=system>)</font>

                            & = & x_i (v)       & - & x_i (u).      \\ \end{array}</math></p>

| &nbsp;

| =

| ''x''_''i''(''u'') + ''x''_''i''(''v'')

| &nbsp;

| =

| ''x''_''i''(''v'') &ndash; ''x''_''i''(''u'').

In the above transcription, the operator bracket of the form "<font face=system>(&nbsp;&hellip;&nbsp;,&nbsp;&hellip;&nbsp;)</font>" is a ''cactus lobe'', signifying ''just one false'', in this case among two boolean variables, while "+" is boolean addition in the proper sense of addition in GF(2), and is thus equivalent to "&ndash;", in the sense of adding the additive inverse.

In the above transcription, the operator bracket of the form "<font face=system>(&nbsp;&hellip;&nbsp;,&nbsp;&hellip;&nbsp;)</font>" is a ''cactus lobe'', signifying ''just one false'', in this case among two boolean variables, while "+" is boolean addition in the proper sense of addition in GF(2), and is thus equivalent to "&ndash;", in the sense of adding the additive inverse.