MyWikiBiz

Changes

User:Jon Awbrey/SCRATCHPAD (view source)

Revision as of 14:40, 4 February 2009

3,165 bytes added ,  14:40, 4 February 2009
more definitions and facts
Line 3,504: Line 3,504:     
D10e. {<o, s> C OxS : <o, s, i> C R for some i C I}
 
D10e. {<o, s> C OxS : <o, s, i> C R for some i C I}
 +
</pre>
 +
 +
===Definition 11===
 +
 +
<pre>
 +
Definition 11
 +
 +
If R c OxSxI,
 +
 +
then the following are identical subsets of SxO:
 +
 +
D11a. RSO
 +
 +
D11b. ROS^
 +
 +
D11c. DenR^
 +
 +
D11d. Den(R)^
 +
 +
D11e. PrSO(R)
 +
 +
D11f. Conv(Den(R))
 +
 +
D11g. {<s, o> C SxO : <o, s, i> C R for some i C I}
 +
</pre>
 +
 +
===Definition 12===
 +
 +
<pre>
 +
Definition 12
 +
 +
If R c OxSxI,
 +
 +
and x C S,
 +
 +
then the following are identical subsets of O:
 +
 +
D12a. ROS.x
 +
 +
D12b. DenR.x
 +
 +
D12c. DenR|x
 +
 +
D12d. DenR(, x)
 +
 +
D12e. Den(R, x)
 +
 +
D12f. Den(R).x
 +
 +
D12g. {o C O : <o, x> C Den(R)}
 +
 +
D12h. {o C O : <o, x, i> C R for some i C I}
 +
</pre>
 +
 +
===Definition 13===
 +
 +
<pre>
 +
Definition 13
 +
 +
If R c OxSxI,
 +
 +
then the following are identical subsets of SxI:
 +
 +
D13a. DerR
 +
 +
D13b. Der(R)
 +
 +
D13c. {<x,y> C SxI : DenR|x = DenR|y}
 +
 +
D13d. {<x,y> C SxI : Den(R, x) = Den(R, y)}
 +
</pre>
 +
 +
===Fact 2.1===
 +
 +
<pre>
 +
Fact 2.1
 +
 +
If R c OxSxI,
 +
 +
then the following are identical subsets of SxI:
 +
 +
F2.1a. DerR :D13a
 +
::
 +
F2.1b. Der(R) :D13b
 +
::
 +
F2.1c. {<x, y> C SxI :
 +
Den(R, x) = Den(R, y)
 +
} :D13c
 +
:R9a
 +
::
 +
F2.1d. {<x, y> C SxI :
 +
{Den(R, x)} = {Den(R, y)}
 +
} :R9b
 +
::
 +
F2.1e. {<x, y> C SxI :
 +
for all o C O
 +
{Den(R, x)}(o) = {Den(R, y)}(o)
 +
} :R9c
 +
::
 +
F2.1f. {<x, y> C SxI :
 +
Conj(o C O)
 +
{Den(R, x)}(o) = {Den(R, y)}(o)
 +
} :R9d
 +
::
 +
F2.1g. {<x, y> C SxI :
 +
Conj(o C O)
 +
(( {Den(R, x)}(o) , {Den(R, y)}(o) ))
 +
} :R9e
 +
::
 +
F2.1h. {<x, y> C SxI :
 +
Conj(o C O)
 +
(( {Den(R, x)} , {Den(R, y)} ))$(o)
 +
} :R9f
 +
:D12e
 +
::
 +
F2.1i. {<x, y> C SxI :
 +
Conj(o C O)
 +
(( {ROS.x} , {ROS.y} ))$(o)
 +
} :D12a
 +
</pre>
 +
 +
===Fact 2.2===
 +
 +
<pre>
 +
Fact 2.2
 +
 +
If R c OxSxI,
 +
 +
then the following are equivalent:
 +
 +
F2.2a. DerR = {<x, y> C SxI :
 +
Conj(o C O)
 +
{Den(R, x)}(o) =
 +
{Den(R, y)}(o)
 +
} :R11a
 +
::
 +
F2.2b. {DerR} = { {<x, y> C SxI :
 +
Conj(o C O)
 +
{Den(R, x)}(o) =
 +
{Den(R, y)}(o)
 +
}
 +
} :R11b
 +
::
 +
F2.2c. {DerR} c SxIxB
 +
:
 +
{DerR} = {<x, y, v> C SxIxB :
 +
v =
 +
[ Conj(o C O)
 +
{Den(R, x)}(o) =
 +
{Den(R, y)}(o)
 +
]
 +
} :R11c
 +
::
 +
F2.2d. {DerR} = {<x, y, v> C SxIxB :
 +
v =
 +
Conj(o C O)
 +
[ {Den(R, x)}(o) =
 +
{Den(R, y)}(o)
 +
]
 +
} :Log
 +
 +
F2.2e. {DerR} = {<x, y, v> C SxIxB :
 +
v =
 +
Conj(o C O)
 +
(( {Den(R, x)}(o),
 +
{Den(R, y)}(o)
 +
))
 +
} :Log
 +
 +
F2.2f. {DerR} = {<x, y, v> C SxIxB :
 +
v =
 +
Conj(o C O)
 +
(( {Den(R, x)},
 +
{Den(R, y)}
 +
))$(o)
 +
} :$
 +
</pre>
 +
 +
===Fact 2.3===
 +
 +
<pre>
 +
Fact 2.3
 +
 +
If R c OxSxI,
 +
 +
then the following are equivalent:
 +
 +
F2.3a. DerR = {<x, y> C SxI :
 +
Conj(o C O)
 +
{Den(R, x)}(o) =
 +
{Den(R, y)}(o)
 +
} :R11a
 +
::
 +
F2.3b. {DerR} : SxI �> B
 +
:
 +
{DerR}(x, y) = [ Conj(o C O)
 +
{Den(R, x)}(o) =
 +
{Den(R, y)}(o)
 +
] :R11d
 +
::
 +
F2.3c. {DerR}(x, y) = Conj(o C O)
 +
[ {Den(R, x)}(o) =
 +
{Den(R, y)}(o)
 +
] :Log
 +
::
 +
F2.3d. {DerR}(x, y) = Conj(o C O)
 +
[ {DenR}(o, x) =
 +
{DenR}(o, y)
 +
] :Def
 +
::
 +
F2.3e. {DerR}(x, y) = Conj(o C O)
 +
(( {DenR}(o, x),
 +
{DenR}(o, y)
 +
)) :Log
 +
:D10b
 +
::
 +
F2.3f. {DerR}(x, y) = Conj(o C O)
 +
(( {ROS}(o, x),
 +
{ROS}(o, y)
 +
)) :D10a
 
</pre>
 
</pre>
Jon Awbrey
12,080

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/78401"