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Revision as of 16:45, 29 April 2009

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→‎Commentary Note 12.2
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===Commentary Note 12.2===
 
===Commentary Note 12.2===
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The term ''exponentiation'' is more generally used in mathematics for operations that involve taking a base to a power, and is slightly preferable to ''involution'' since the latter is used for different concepts in different contexts.  Operations analogous to taking powers are widespread throughout mathematics and Peirce frequently makes use of them in a number of important applications, for example, in his theory of information.  But that's another story.
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{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
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<p>Then</p>
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| align="center" | <math>(\mathit{s}^\mathit{l})^\mathrm{w}\!</math>
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<p>will denote whatever stands to every woman in the relation of servant of every lover of hers;</p>
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The ''function space'' <math>Y^X,\!</math> where <math>X\!</math> and <math>Y\!</math> are sets, is the set of all functions from <math>X\!</math> to <math>Y.\!</math> An alternative notation for <math>Y^X\!</math> is <math>(X \to Y).</math> Thus we have the following equivalents:
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<p>and</p>
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| align="center" | <math>\mathit{s}^{(\mathit{l}\mathrm{w})}\!</math>
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<p>will denote whatever is a servant of everything that is lover of a woman.</p>
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{| align="center" cellspacing="6" width="90%"
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<p>So that</p>
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| <math>\begin{matrix}Y^X & = & (X \to Y) & = & \{ f : X \to Y \}\end{matrix}</math>
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| align="center" | <math>(\mathit{s}^\mathit{l})^\mathrm{w} ~=~ \mathit{s}^{(\mathit{l}\mathrm{w})}.</math>
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If <math>X\!</math> and <math>Y\!</math> have cardinalities <math>|X|\!</math> and <math>|Y|,\!</math> respectively, then the function space <math>Y^X\!</math> has a cardinality given by the following equation:
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<p>(Peirce, CP 3.77).</p>
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{| align="center" cellspacing="6" width="90%"
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| <math>\begin{matrix}|Y^X| & = & |Y|^{|X|}\end{matrix}</math>
   
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Jon Awbrey
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