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User:Jon Awbrey/SCRATCHPAD (view source)

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<br>
 
<br>
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==Work Area==
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<pre>
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Logical Translation Rule 1
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If S is a sentence
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about things in the universe U
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and P is a proposition : U -> B, such that:
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L1a. [S]  =  P,
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then the following equations hold:
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L1b00. [False] = () = 0 : U->B.
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L1b01. [Not S] = ([S]) = (P) : U->B.
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L1b10. [S] = [S] = P : U->B.
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L1b11. [True] = (()) = 1 : U->B.
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</pre>
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<br>
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<pre>
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Geometric Translation Rule 1
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If X c U
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and P : U -> B, such that:
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G1a. {X}  =  P,
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then the following equations hold:
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G1b00. {{}} = () = 0 : U->B.
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G1b10. {~X} = ({X}) = (P) : U->B.
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G1b01. {X} = {X} = P : U->B.
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G1b11. {U} = (()) = 1 : U->B.
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</pre>
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<br>
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<pre>
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Logical Translation Rule 2
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If S, T are sentences
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about things in the universe U
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and P, Q are propositions: U -> B, such that:
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L2a. [S] = P  and  [T] = Q,
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then the following equations hold:
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L2b00. [False] = () = 0 : U->B.
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L2b01. [Neither S nor T] = ([S])([T]) = (P)(Q).
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L2b02. [Not S, but T] = ([S])[T] = (P) Q.
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L2b03. [Not S] = ([S]) = (P).
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L2b04. [S and not T] = [S]([T]) = P (Q).
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L2b05. [Not T] = ([T]) = (Q).
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L2b06. [S or T, not both] = ([S], [T]) = (P, Q).
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L2b07. [Not both S and T] = ([S].[T]) = (P Q).
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L2b08. [S and T] = [S].[T] = P.Q.
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L2b09. [S <=> T] = (([S], [T])) = ((P, Q)).
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L2b10. [T] = [T] = Q.
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L2b11. [S => T] = ([S]([T])) = (P (Q)).
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L2b12. [S] = [S] = P.
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L2b13. [S <= T] = (([S]) [T]) = ((P) Q).
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L2b14. [S or T] = (([S])([T])) = ((P)(Q)).
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L2b15. [True] = (()) = 1 : U->B.
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</pre>
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<br>
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<pre>
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Geometric Translation Rule 2
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If X, Y c U
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and P, Q U -> B, such that:
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G2a. {X} = P  and  {Y} = Q,
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then the following equations hold:
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G2b00. {{}} = () = 0 : U->B.
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G2b01. {~X n ~Y} = ({X})({Y}) = (P)(Q).
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G2b02. {~X n Y} = ({X}){Y} = (P) Q.
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G2b03. {~X} = ({X}) = (P).
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G2b04. {X n ~Y} = {X}({Y}) = P (Q).
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G2b05. {~Y} = ({Y}) = (Q).
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G2b06. {X + Y} = ({X}, {Y}) = (P, Q).
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G2b07. {~(X n Y)} = ({X}.{Y}) = (P Q).
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G2b08. {X n Y} = {X}.{Y} = P.Q.
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G2b09. {~(X + Y)} = (({X}, {Y})) = ((P, Q)).
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G2b10. {Y} = {Y} = Q.
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G2b11. {~(X n ~Y)} = ({X}({Y})) = (P (Q)).
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G2b12. {X} = {X} = P.
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G2b13. {~(~X n Y)} = (({X}) {Y}) = ((P) Q).
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G2b14. {X u Y} = (({X})({Y})) = ((P)(Q)).
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G2b15. {U} = (()) = 1 : U->B.
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</pre>
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<br>
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<pre>
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Value Rule 1
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If v, w C B
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then "v = w" is a sentence about <v, w> C B2,
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[v = w] is a proposition : B2 -> B,
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and the following are identical values in B:
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V1a. [ v = w ](v, w)
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V1b. [ v <=> w ](v, w)
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V1c. ((v , w))
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</pre>
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<br>
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<pre>
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Value Rule 1
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If v, w C B,
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then the following are equivalent:
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V1a. v = w.
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V1b. v <=> w.
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V1c. (( v , w )).
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</pre>
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<pre>
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A rule that allows one to turn equivalent sentences into identical propositions:
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(S <=> T) <=> ([S] = [T])
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Consider [ v = w ](v, w) and [ v(u) = w(u) ](u)
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Value Rule 1
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If v, w C B,
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then the following are identical values in B:
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V1a. [ v = w ]
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V1b. [ v <=> w ]
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V1c. (( v , w ))
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Value Rule 1
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If f, g : U -> B,
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and u C U
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then the following are identical values in B:
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V1a. [ f(u) = g(u) ]
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V1b. [ f(u) <=> g(u) ]
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V1c. (( f(u) , g(u) ))
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Value Rule 1
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If f, g : U -> B,
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then the following are identical propositions on U:
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V1a. [ f = g ]
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V1b. [ f <=> g ]
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V1c. (( f , g ))$
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Evaluation Rule 1
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If f, g : U -> B
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and u C U,
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then the following are equivalent:
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E1a. f(u) = g(u). :V1a
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::
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E1b. f(u) <=> g(u). :V1b
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::
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E1c. (( f(u) , g(u) )). :V1c
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:$1a
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::
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E1d. (( f , g ))$(u). :$1b
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Evaluation Rule 1
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If S, T are sentences
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about things in the universe U,
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f, g are propositions: U -> B,
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and u C U,
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then the following are equivalent:
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E1a. f(u) = g(u). :V1a
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::
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E1b. f(u) <=> g(u). :V1b
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::
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E1c. (( f(u) , g(u) )). :V1c
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:$1a
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::
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E1d. (( f , g ))$(u). :$1b
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Definition 2
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If X, Y c U,
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then the following are equivalent:
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D2a. X = Y.
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D2b. u C X  <=>  u C Y, for all u C U.
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Definition 3
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If f, g : U -> V,
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then the following are equivalent:
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D3a. f = g.
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D3b. f(u) = g(u), for all u C U.
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Definition 4
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If X c U,
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then the following are identical subsets of UxB:
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D4a. {X}
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D4b. {<u, v> C UxB : v = [u C X]}
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Definition 5
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If X c U,
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then the following are identical propositions:
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D5a. {X}.
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D5b. f : U -> B
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: f(u) = [u C X], for all u C U.
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Given an indexed set of sentences, Sj for j C J, it is possible to consider the logical conjunction of the corresponding propositions.  Various notations for this concept are be useful in various contexts, a sufficient sample of which are recorded in Definition 6.
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Definition 6
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If Sj is a sentence
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about things in the universe U,
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for all j C J,
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then the following are equivalent:
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D6a. Sj, for all j C J.
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D6b. For all j C J, Sj.
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D6c. Conj(j C J) Sj.
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D6d. ConjJ,j Sj.
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D6e. ConjJj Sj.
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Definition 7
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If S, T are sentences
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about things in the universe U,
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then the following are equivalent:
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D7a. S <=> T.
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D7b. [S] = [T].
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Rule 5
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If X, Y c U,
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then the following are equivalent:
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R5a. X = Y. :D2a
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::
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R5b. u C X  <=>  u C Y, for all u C U. :D2b
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:D7a
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::
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R5c. [u C X] = [u C Y], for all u C U. :D7b
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:???
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::
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R5d. {<u, v> C UxB : v = [u C X]}
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=
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{<u, v> C UxB : v = [u C Y]}. :???
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:D5b
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::
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R5e. {X} = {Y}. :D5a
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Rule 6
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If f, g : U -> V,
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then the following are equivalent:
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R6a. f = g. :D3a
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::
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R6b. f(u) = g(u), for all u C U. :D3b
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:D6a
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::
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R6c. ConjUu (f(u) = g(u)). :D6e
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Rule 7
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If P, Q : U -> B,
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then the following are equivalent:
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R7a. P = Q. :R6a
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::
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R7b. P(u) = Q(u), for all u C U. :R6b
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::
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R7c. ConjUu (P(u)  =  Q(u)). :R6c
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:P1a
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::
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R7d. ConjUu (P(u) <=> Q(u)). :P1b
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::
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R7e. ConjUu (( P(u) , Q(u) )). :P1c
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:$1a
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::
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R7f. ConjUu (( P , Q ))$(u). :$1b
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Rule 8
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If S, T are sentences
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about things in the universe U,
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then the following are equivalent:
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R8a. S <=> T. :D7a
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::
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R8b. [S] = [T]. :D7b
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:R7a
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::
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R8c. [S](u) = [T](u), for all u C U. :R7b
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::
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R8d. ConjUu ( [S](u)  =  [T](u) ). :R7c
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::
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R8e. ConjUu ( [S](u) <=> [T](u) ). :R7d
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::
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R8f. ConjUu (( [S](u) , [T](u) )). :R7e
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::
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R8g. ConjUu (( [S] , [T] ))$(u). :R7f
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For instance, the observation that expresses the equality of sets in terms of their indicator functions can be formalized according to the pattern in Rule 9, namely, at lines (a, b, c), and these components of Rule 9 can be cited in future uses as "R9a", "R9b", "R9c", respectively.  Using Rule 7, annotated as "R7", to adduce a few properties of indicator functions to the account, it is possible to extend Rule 9 by another few steps, referenced as "R9d", "R9e", "R9f", "R9g".
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Rule 9
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If X, Y c U,
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then the following are equivalent:
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R9a. X = Y. :R5a
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::
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R9b. {X} = {Y}. :R5e
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:R7a
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::
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R9c. {X}(u) = {Y}(u), for all u C U. :R7b
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::
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R9d. ConjUu ( {X}(u)  =  {Y}(u) ). :R7c
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::
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R9e. ConjUu ( {X}(u) <=> {Y}(u) ). :R7d
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::
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R9f. ConjUu (( {X}(u) , {Y}(u) )). :R7e
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::
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R9g. ConjUu (( {X} , {Y} ))$(u). :R7f
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Rule 10
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If X, Y c U,
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then the following are equivalent:
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R10a. X = Y. :D2a
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::
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R10b. u C X  <=>  u C Y, for all u C U. :D2b
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:R8a
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::
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R10c. [u C X] = [u C Y]. :R8b
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::
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R10d. For all u C U,
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[u C X](u) = [u C Y](u). :R8c
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::
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R10e. ConjUu ( [u C X](u)  =  [u C Y](u) ). :R8d
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::
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R10f. ConjUu ( [u C X](u) <=> [u C Y](u) ). :R8e
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::
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R10g. ConjUu (( [u C X](u) , [u C Y](u) )). :R8f
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::
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R10h. ConjUu (( [u C X] , [u C Y] ))$(u). :R8g
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Rule 11
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If X c U
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then the following are equivalent:
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R11a. X = {u C U : S}. :R5a
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::
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R11b. {X} = { {u C U : S} }. :R5e
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::
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R11c. {X} c UxB
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: {X} = {<u, v> C UxB : v = [S](u)}. :R
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::
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R11d. {X} : U -> B
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: {X}(u) = [S](u), for all u C U. :R
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::
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R11e. {X} = [S]. :R
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</pre>
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<pre>
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Fact 1
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If X,Y c U,
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then the following are equivalent:
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F1a. S <=> X = Y. :R9a
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::
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F1b. S <=> {X} = {Y}. :R9b
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::
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F1c. S <=> {X}(u) = {Y}(u), for all u C U. :R9c
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::
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F1d. S <=> ConjUu ( {X}(u) = {Y}(u) ). :R9d
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:R8a
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::
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F1e. [S] = [ ConjUu ( {X}(u) = {Y}(u) ) ]. :R8b
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:???
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::
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F1f. [S] = ConjUu [ {X}(u) = {Y}(u) ]. :???
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::
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F1g. [S] = ConjUu (( {X}(u) , {Y}(u) )). :$1a
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::
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F1h. [S] = ConjUu (( {X} , {Y} ))$(u). :$1b
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///
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{u C U : (f, g)$(u)}
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= {u C U : (f(u), g(u))}
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= {u C
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///
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</pre>
Jon Awbrey
12,080

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