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| | </pre> | | </pre> |
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| − | =====1.3.12.2. Derived Equivalence Relations===== | + | =====1.3.12.2. Derived Equivalence Relations <big>✔</big>===== |
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| − | The uses of the equal sign for denoting equations or equivalences are recalled and extended in the following ways:
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| − | 1. If E is an arbitrary equivalence relation,
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| − | then the equation "x =E y" means that <x, y> C E.
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| − | 2. If R is a sign relation such that RSI is a SER on S = I,
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| − | then the semiotic equation "x =R y" means that <x, y> C RSI.
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| − | 3. If R is a sign relation such that F is its DER on S = I,
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| − | then the denotative equation "x =R y" means that <x, y> C F,
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| − | in other words, that Den(R, x) = Den(R, y).
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| − | The uses of square brackets for denoting equivalence classes are recalled and extended in the following ways:
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| − | 1. If E is an arbitrary equivalence relation,
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| − | then "[x]E" denotes the equivalence class of x under E.
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| − | 2. If R is a sign relation such that Con(R) is a SER on S = I,
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| − | then "[x]R" denotes the SEC of x under Con(R).
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| − | 3. If R is a sign relation such that Der(R) is a DER on S = I,
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| − | then "[x]R" denotes the DEC of x under Der(R).
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| − | By applying the form of Fact 1 to the special case where X = Den(R, x) and Y = Den(R, y), one obtains the following facts.
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| − | <pre>
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| − | Fact 2.1
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| − | If R c OxSxI,
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| − | then the following are identical subsets of SxI:
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| − | F2.1a. DerR :D13a
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| − | ::
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| − | F2.1b. Der(R) :D13b
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| − | ::
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| − | F2.1c. {<x, y> C SxI :
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| − | Den(R, x) = Den(R, y)
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| − | } :D13c
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| − | :R9a
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| − | ::
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| − | F2.1d. {<x, y> C SxI :
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| − | {Den(R, x)} = {Den(R, y)}
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| − | } :R9b
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| − | | |
| − | ::
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| − | F2.1e. {<x, y> C SxI :
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| − | for all o C O
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| − | {Den(R, x)}(o) = {Den(R, y)}(o)
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| − | | |
| − | } :R9c
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| − | | |
| − | ::
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| − | F2.1f. {<x, y> C SxI :
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| − | Conj(o C O)
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| − | {Den(R, x)}(o) = {Den(R, y)}(o)
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| − | | |
| − | } :R9d
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| − | ::
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| − | F2.1g. {<x, y> C SxI :
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| − | Conj(o C O)
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| − | (( {Den(R, x)}(o) , {Den(R, y)}(o) ))
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| − | } :R9e
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| − | ::
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| − | F2.1h. {<x, y> C SxI :
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| − | Conj(o C O)
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| − | (( {Den(R, x)} , {Den(R, y)} ))$(o)
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| − | } :R9f
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| − | :D12e
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| − | ::
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| − | F2.1i. {<x, y> C SxI :
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| − | Conj(o C O)
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| − | (( {ROS.x} , {ROS.y} ))$(o)
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| − | } :D12a
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| − | </pre> | |
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| − | <pre>
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| − | Fact 2.2
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| − | If R c OxSxI,
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| − | then the following are equivalent:
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| − | F2.2a. DerR = {<x, y> C SxI :
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| − | Conj(o C O)
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| − | {Den(R, x)}(o) =
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| − | {Den(R, y)}(o)
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| − | } :R11a
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| − | ::
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| − | F2.2b. {DerR} = { {<x, y> C SxI :
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| − | Conj(o C O)
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| − | {Den(R, x)}(o) =
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| − | {Den(R, y)}(o)
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| − | }
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| − | } :R11b
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| − | ::
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| − | F2.2c. {DerR} c SxIxB
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| − | :
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| − | {DerR} = {<x, y, v> C SxIxB :
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| − | v =
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| − | [ Conj(o C O)
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| − | {Den(R, x)}(o) =
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| − | {Den(R, y)}(o)
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| − | ]
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| − | } :R11c
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| − | ::
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| − | F2.2d. {DerR} = {<x, y, v> C SxIxB :
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| − | v =
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| − | Conj(o C O)
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| − | [ {Den(R, x)}(o) =
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| − | {Den(R, y)}(o)
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| − | ]
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| − | } :Log
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| − | F2.2e. {DerR} = {<x, y, v> C SxIxB :
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| − | v =
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| − | Conj(o C O)
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| − | (( {Den(R, x)}(o),
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| − | {Den(R, y)}(o)
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| − | ))
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| − | } :Log
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| − | F2.2f. {DerR} = {<x, y, v> C SxIxB :
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| − | v =
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| − | Conj(o C O)
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| − | (( {Den(R, x)},
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| − | {Den(R, y)}
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| − | ))$(o)
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| − | } :$
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| − | </pre>
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| − | | |
| − | <pre>
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| − | Fact 2.3
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| − | If R c OxSxI,
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| − | then the following are equivalent:
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| − | F2.3a. DerR = {<x, y> C SxI :
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| − | Conj(o C O)
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| − | {Den(R, x)}(o) =
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| − | {Den(R, y)}(o)
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| − | } :R11a
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| − | ::
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| − | F2.3b. {DerR} : SxI -> B
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| − | :
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| − | {DerR}(x, y) = [ Conj(o C O)
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| − | {Den(R, x)}(o) =
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| − | {Den(R, y)}(o)
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| − | ] :R11d
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| − | ::
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| − | F2.3c. {DerR}(x, y) = Conj(o C O)
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| − | | |
| − | [ {Den(R, x)}(o) =
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| − | {Den(R, y)}(o)
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| − | ] :Log
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| − | | |
| − | ::
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| − | F2.3d. {DerR}(x, y) = Conj(o C O)
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| − | [ {DenR}(o, x) =
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| − | {DenR}(o, y)
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| − | ] :Def
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| − | | |
| − | ::
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| − | F2.3e. {DerR}(x, y) = Conj(o C O)
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| − | (( {DenR}(o, x),
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| − | {DenR}(o, y)
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| − | )) :Log
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| − | :D10b
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| − | ::
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| − | F2.3f. {DerR}(x, y) = Conj(o C O)
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| − | (( {ROS}(o, x),
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| − | {ROS}(o, y)
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| − | )) :D10a
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| − | </pre>
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| | =====1.3.12.3. Digression on Derived Relations <big>✔</big>===== | | =====1.3.12.3. Digression on Derived Relations <big>✔</big>===== |
