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Revision as of 19:34, 23 June 2008
, 19:34, 23 June 2008→Special Classes of Propositions: use "text binomial" <math>\tbinom{n}{k}</math> for binomial coefficient
Among the <math>2^{2^n}</math> propositions or functions in ('''B'''<sup>''n''</sup> → '''B''') are several fundamental sets of 2<sup>''n''</sup> propositions each that take on special forms with respect to a given basis <font face="lucida calligraphy">A</font> = {''a''<sub>''i''</sub>}. Three of these forms are especially common, the ''linear'', the ''positive'', and the ''singular''
Among the <math>2^{2^n}</math> propositions or functions in ('''B'''<sup>''n''</sup> → '''B''') are several fundamental sets of 2<sup>''n''</sup> propositions each that take on special forms with respect to a given basis <font face="lucida calligraphy">A</font> = {''a''<sub>''i''</sub>}. Three of these forms are especially common, the ''linear'', the ''positive'', and the ''singular''
propositions. Each set is naturally parameterized by the coordinate vectors in '''B'''<sup>''n''</sup> and falls into ''n''+1 ranks, with a binomial coefficient ''C''(''n'', ''k'') giving the number of propositions that have rank or weight ''k''.
propositions. Each set is naturally parameterized by the coordinate vectors in '''B'''<sup>''n''</sup> and falls into ''n''+1 ranks, with a binomial coefficient <math>\tbinom{n}{k}</math> giving the number of propositions that have rank or weight ''k''.
The ''linear propositions'', {hom : '''B'''<sup>''n''</sup> → '''B'''} = ('''B'''<sup>''n''</sup> <font face=symbol>'''+>'''</font> '''B'''), may be expressed as sums of the following form:
The ''linear propositions'', {hom : '''B'''<sup>''n''</sup> → '''B'''} = ('''B'''<sup>''n''</sup> <font face=symbol>'''+>'''</font> '''B'''), may be expressed as sums of the following form:
