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| | ======Other Rules====== | | ======Other Rules====== |
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| − | <pre>
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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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| − | 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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| − | 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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| − | </pre>
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| − | <pre>
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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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| − | R6b. f(u) = g(u), for all u C U. :D3b
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| − | :D6a
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| − | R6c. ConjUu (f(u) = g(u)). :D6e
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| − | </pre>
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| − | <pre>
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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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| − | R7b. P(u) = Q(u), for all u C U. :R6b
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| − | R7c. ConjUu (P(u) = Q(u)). :R6c
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| − | :P1a
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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). :$1b
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| − | </pre>
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| − | <pre>
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| − | Rule 8
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| − | If S, T are sentences
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| − | about things in the universe U,
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| − | then the following are equivalent:
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| − | R8a. S <=> T. :D7a
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| − | ::
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| − | R8b. [S] = [T]. :D7b
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| − | :R7a
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| − | R8c. [S](u) = [T](u), for all u C U. :R7b
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| − | ::
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| − | R8d. ConjUu ( [S](u) = [T](u) ). :R7c
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| − | R8e. ConjUu ( [S](u) <=> [T](u) ). :R7d
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| − | R8f. ConjUu (( [S](u) , [T](u) )). :R7e
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| − | R8g. ConjUu (( [S] , [T] ))$(u). :R7f
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| − | </pre>
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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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| − |
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| − | <pre>
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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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| − | 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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| − | 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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| − | </pre>
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| | <pre> | | <pre> |
