MyWikiBiz, Author Your Legacy — Wednesday September 10, 2025
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, 02:42, 11 September 2010
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| | : f(u) = [u C X], for all u C U. | | : f(u) = [u C X], for all u C U. |
| | </pre> | | </pre> |
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| − | ======Other Rules======
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| − | <pre>
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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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| − | </pre>
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| − | <pre>
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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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| − | An application of Rule 11 involves the recognition of an antecedent condition as a case under the Rule, that is, as a condition that matches one of the sentences in the Rule's chain of equivalents, and it requires the relegation of the other expressions to the production of a result. Thus, there is the choice of an initial expression that has to be checked on input for whether it fits the antecedent condition, and there is the choice of three types of output that are generated as a consequence, only one of which is generally needed at any given time. More often than not, though, a rule is applied in only a few of its possible ways. The usual antecedent and the usual consequents for Rule 11 can be distinguished in form and specialized in practice as follows:
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| − | a. R11a marks the usual starting place for an application of the Rule, that is, the standard form of antecedent condition that is likely to lead to an invocation of the Rule.
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| − | b. R11b records the trivial consequence of applying the spiny braces to both sides of the initial equation.
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| − | c. R11c gives a version of the indicator function with {X} c UxB, called its "extensional form".
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| − | d. R11d gives a version of the indicator function with {X} : U->B, called its "functional form".
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| | ======Facts====== | | ======Facts====== |
