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Revision as of 01:59, 27 May 2008
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: Let <math>U = X_1 \times \ldots \times X_k</math> and <math>\operatorname{E}U = U \times \operatorname{d}U = X_1 \times \ldots \times X_k \times \operatorname{d}X_1 \times \ldots \times \operatorname{d}X_k.</math>
: Let <math>U = X_1 \times \ldots \times X_k</math> and <math>\operatorname{E}U = U \times \operatorname{d}U = X_1 \times \ldots \times X_k \times \operatorname{d}X_1 \times \ldots \times \operatorname{d}X_k.</math>
For a proposition ''f'' : ''X''<sub>1</sub> × … × ''X''<sub>''k''</sub> → '''B''', the (first order) enlargement of f is the proposition ''Ef'' : ''EU'' → '''B''' that is defined by:
For a proposition <math>f : X_1 \times \ldots \times X_k \to \mathbb{B},</math> the (first order) enlargement of <math>f\!</math> is the proposition <math>\operatorname{E}f : \operatorname{E}U \to \mathbb{B}</math> that is defined by:
: ''Ef''(''x''<sub>1</sub>, …, ''x''<sub>''k''</sub>, ''dx''<sub>1</sub>, …, ''dx''<sub>''k''</sub>) = ''f''(''x''<sub>1</sub> + ''dx''<sub>1</sub>, …, ''x''<sub>''k''</sub> + ''dx''<sub>''k''</sub>).
: <p><math>\operatorname{E}f(x_1, \ldots x_k, \operatorname{d}x_1, \ldots, \operatorname{d}x_k) = f(x_1 + \operatorname{d}x_1, \ldots, x_k + \operatorname{d}x_k).</math></p>
It should be noted that the so-called ''differential variables'' ''dx''<sub>''j''</sub> are really just the same kind of boolean variables as the other ''x''<sub>''j''</sub>. It is conventional to give the additional variables these brands of inflected names, but whatever extra connotations we might choose to attach to these syntactic conveniences are wholly external to their purely algebraic meanings.
It should be noted that the so-called ''differential variables'' ''dx''<sub>''j''</sub> are really just the same kind of boolean variables as the other ''x''<sub>''j''</sub>. It is conventional to give the additional variables these brands of inflected names, but whatever extra connotations we might choose to attach to these syntactic conveniences are wholly external to their purely algebraic meanings.