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Revision as of 19:32, 9 March 2009
, 19:32, 9 March 2009→Note 13: \texttt
A sketch of this work is presented in the following series of Figures, where each logical proposition is expanded over the basic cells <math>\texttt{uv}, \texttt{u(v)}, \texttt{(u)v}, \texttt{(u)(v)}</math> of the 2-dimensional universe of discourse <math>U^\circ = [u, v].\!</math>
A sketch of this work is presented in the following series of Figures, where each logical proposition is expanded over the basic cells <math>\texttt{uv}, \texttt{u(v)}, \texttt{(u)v}, \texttt{(u)(v)}</math> of the 2-dimensional universe of discourse <math>U^\circ = [u, v].\!</math>
===Computation Summary for <math>f(u, v) = \texttt{((u)(v))}</math>===
Computation Summary for f<u, v> = ((u)(v))
Figure 1.1 expands f = ((u)(v)) over [u, v] to produce
Figure 1.1 is a venn diagram that show how the proposition <math>f = \texttt{((u)(v))}</math> can be expanded over the universe of discourse <math>[u, v]\!</math> to produce a logically equivalent exclusive disjunction, namely, <math>\texttt{uv~+~u(v)~+~(u)v}.</math>
<pre>
o---------------------------------------o
o---------------------------------------o
|```````````````````````````````````````|
|```````````````````````````````````````|
o---------------------------------------o
o---------------------------------------o
Figure 1.1. f = ((u)(v))
Figure 1.1. f = ((u)(v))
</pre>
Figure 1.2 expands Ef = ((u + du)(v + dv)) over [u, v] to give:
Figure 1.2 expands Ef = ((u + du)(v + dv)) over [u, v] to give:
uv.(du dv) + u(v).(du (dv)) + (u)v.((du) dv) + (u)(v).((du)(dv))
uv.(du dv) + u(v).(du (dv)) + (u)v.((du) dv) + (u)(v).((du)(dv))
<pre>
o---------------------------------------o
o---------------------------------------o
|```````````````````````````````````````|
|```````````````````````````````````````|
o---------------------------------------o
o---------------------------------------o
Figure 1.2. Ef = ((u + du)(v + dv))
Figure 1.2. Ef = ((u + du)(v + dv))
</pre>
Figure 1.3 expands Df = f + Ef over [u, v] to produce:
Figure 1.3 expands Df = f + Ef over [u, v] to produce:
uv.du dv + u(v).du(dv) + (u)v.(du)dv + (u)(v).((du)(dv))
uv.du dv + u(v).du(dv) + (u)v.(du)dv + (u)(v).((du)(dv))
<pre>
o---------------------------------------o
o---------------------------------------o
|```````````````````````````````````````|
|```````````````````````````````````````|
o---------------------------------------o
o---------------------------------------o
Figure 1.3. Df = f + Ef
Figure 1.3. Df = f + Ef
</pre>
I'll break this here in case anyone wants
I'll break this here in case anyone wants to try and do the work for <math>g\!</math> on their own.
to try and do the work for g on their own.
==Note 14==
==Note 14==
