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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
Revision as of 20:18, 19 November 2012
, 20:18, 19 November 2012→6.32. Partiality : Selective Operations: markup
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<math>\begin{array}{lll}
<math>\begin{array}{lll}
A^\sharp : X \to \mathbb{B} & \text{where} & A^\sharp (x) = 1 \iff x \in A.
A^\sharp : X \to \mathbb{B} & \text{such that} & A^\sharp (x) = 1 \iff x \in A.
\end{array}</math>
\end{array}</math>
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Other names for the same concept, appearing under various notations, are the ''characteristic function'' or the ''indicator function'' of <math>A\!</math> in <math>X\!</math>.
Other names for the same concept, appearing under various notations, are the ''characteristic function'' or the ''indicator function'' of <math>A\!</math> in <math>X.\!</math>
Conversely, given a boolean-valued function <math>f : X \to \mathbb{B},\!</math> let the ''selected set of <math>f\!</math> in <math>X\!</math>'' be notated as <math>f_\flat\!</math> and defined as follows.
Conversely, if one has a binary valued function f : X > B, then "f#", read as "f numbd" or "f selection", denotes the "selected set" of f, defined as:
{| align="center" cellspacing="8" width="90%"
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<math>\begin{array}{lll}
f_\flat \subseteq X & \text{such that} & f_\flat = f^{-1}(1) = \{ x \in X : f(x) = 1 \}.
\end{array}</math>
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Other names for this subset are the "fiber", "pre image", "level set", or "antecedents" of 1 under the mapping f.
Other names for the same concept are the ''fiber'', ''level set'', or ''pre-image'' of 1 under the mapping <math>f : X \to \mathbb{B}.\!</math>
<pre>
Obviously, the relation between these operations is such that:
Obviously, the relation between these operations is such that: