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User:Jon Awbrey/SCRATCHPAD (view source)
Revision as of 20:51, 29 January 2009
, 20:51, 29 January 2009→Evaluation Rule 1
===Evaluation Rule 1===
===Evaluation Rule 1===
====Variant 1====
<pre>
<pre>
If f, g : U -> B
If f, g : U -> B
and u C U,
and u C U,
{| align="center" cellpadding="0" cellspacing="0" width="100%"
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:48px; text-align:right"
|- style="height:48px; text-align:right"
| width="98%" | <math>\text{Value Rule 1}\!</math>
| width="98%" | <math>\text{Evaluation Rule 1}\!</math>
| width="2%" |
| width="2%" |
|}
|}
| width="14%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="14%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="84%" style="border-top:1px solid black" | <math>f, g ~:~ X \to \underline\mathbb{B}</math>
| width="84%" style="border-top:1px solid black" | <math>f, g ~:~ X \to \underline\mathbb{B}</math>
|- style="height:48px"
|
| <math>\text{and}\!</math>
| <math>x ~\in~ X</math>
|- style="height:48px"
|- style="height:48px"
|
|
| <math>\text{then}\!</math>
| <math>\text{then}\!</math>
| <math>\text{the following are identical propositions on}~ X:</math>
| <math>\text{the following are equivalent:}\!</math>
|}
|}
|-
|-
|- style="height:56px"
|- style="height:56px"
| width="2%" style="border-top:1px solid black" |
| width="2%" style="border-top:1px solid black" |
| width="14%" style="border-top:1px solid black" | <math>\text{V1a.}\!</math>
| width="14%" style="border-top:1px solid black" | <math>\text{E1a.}\!</math>
| width="84%" style="border-top:1px solid black" | <math>\downharpoonleft f ~=~ g \downharpoonright</math>
| width="84%" style="border-top:1px solid black" | <math>f(x) ~=~ g(x)</math>
|- style="height:56px"
|
| <math>\text{E1b.}\!</math>
| <math>f(x) ~\Leftrightarrow~ g(x)</math>
|- style="height:56px"
|- style="height:56px"
|
|
| <math>\text{V1b.}\!</math>
| <math>\text{E1c.}\!</math>
| <math>\downharpoonleft f ~\Leftrightarrow~ g \downharpoonright</math>
| <math>\underline{((}~ f(x) ~,~ g(x) ~\underline{))}</math>
|- style="height:56px"
|- style="height:56px"
|
|
| <math>\text{V1c.}\!</math>
| <math>\text{E1d.}\!</math>
| <math>\underline{((}~ f ~,~ g ~\underline{))}^\$</math>
| <math>\underline{((}~ f ~,~ g ~\underline{))}^\$ (x)</math>
|}
|}
|}
|}
<br>
<br>
====Variant 2====
<pre>
Evaluation Rule 1
If S, T are sentences
about things in the universe U,
f, g are propositions: U -> B,
and u C U,
then the following are equivalent:
E1a. f(u) = g(u). :V1a
::
E1b. f(u) <=> g(u). :V1b
::
E1c. (( f(u) , g(u) )). :V1c
:$1a
::
E1d. (( f , g ))$(u). :$1b
</pre>
===Definition 2===
===Definition 2===