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| 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> |
| | | |
− | <pre>
| + | ===Computation Summary for <math>f(u, v) = \texttt{((u)(v))}</math>=== |
− | Computation Summary for f<u, v> = ((u)(v)) | |
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− | 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> |
− | the equivalent exclusive disjunction uv + u(v) + (u)v.
| |
| | | |
| + | <pre> |
| o---------------------------------------o | | o---------------------------------------o |
| |```````````````````````````````````````| | | |```````````````````````````````````````| |
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| o---------------------------------------o | | o---------------------------------------o |
| Figure 1.1. f = ((u)(v)) | | Figure 1.1. f = ((u)(v)) |
| + | </pre> |
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− | 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: |
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| 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 |
| |```````````````````````````````````````| | | |```````````````````````````````````````| |
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| o---------------------------------------o | | o---------------------------------------o |
| Figure 1.2. Ef = ((u + du)(v + dv)) | | Figure 1.2. Ef = ((u + du)(v + dv)) |
| + | </pre> |
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| Figure 1.3 expands Df = f + Ef over [u, v] to produce: | | Figure 1.3 expands Df = f + Ef over [u, v] to produce: |
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| 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)) |
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| + | <pre> |
| o---------------------------------------o | | o---------------------------------------o |
| |```````````````````````````````````````| | | |```````````````````````````````````````| |
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| o---------------------------------------o | | o---------------------------------------o |
| Figure 1.3. Df = f + Ef | | Figure 1.3. Df = f + Ef |
| + | </pre> |
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− | 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. | |
− | </pre>
| |
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| ==Note 14== | | ==Note 14== |