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| | ======Definitions====== | | ======Definitions====== |
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| − | <pre>
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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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| − | </pre>
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| − | <pre>
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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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| − | </pre>
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| − | <pre>
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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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| − | </pre>
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| | <pre> | | <pre> |
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| | </pre> | | </pre> |
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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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| − | <pre>
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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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| − | </pre>
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| | <pre> | | <pre> |
| − | 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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| − | </pre>
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| | ======Other Rules====== | | ======Other Rules====== |