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Directory:Jon Awbrey/Papers/Inquiry Driven Systems (view source)
Revision as of 19:27, 12 January 2009
, 19:27, 12 January 2009→1.3.10.3. Propositions and Sentences: markup
<p>Considered in extensional form, <math>f_Q\!</math> is the subset of <math>X \times \underline\mathbb{B}</math> that is given by the following formula:</p>
<p>Considered in extensional form, <math>f_Q\!</math> is the subset of <math>X \times \underline\mathbb{B}</math> that is given by the following formula:</p>
<p><math>f_Q \ = \ \{ (x, b) \in X \times \underline\mathbb{B} \ : \ b = \underline{1} \ \Leftrightarrow \ x \in Q \}.</math></p></li>
<p><math>f_Q ~=~ \{ (x, b) \in X \times \underline\mathbb{B} ~:~ b = \underline{1} ~\Leftrightarrow~ x \in Q \}.</math></p></li>
<li>
<li>
<p>Considered in functional form, <math>f_Q\!</math> is the map from <math>X\!</math> to <math>\underline\mathbb{B}</math> that is given by the following condition:</p>
<p>Considered in functional form, <math>f_Q\!</math> is the map from <math>X\!</math> to <math>\underline\mathbb{B}</math> that is given by the following condition:</p>
<p><math>f_Q (x) = \underline{1} \ \Leftrightarrow \ x \in Q.</math></p></li>
<p><math>f_Q (x) = \underline{1} ~\Leftrightarrow~ x \in Q.</math></p></li>
</ol>
</ol>
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
| <math>\operatorname{Fiber~of}~ y ~\operatorname{under}~ f \ = \ f^{-1} (y) \ = \ \{ x \in X : f(x) = y \}.</math>
| <math>\operatorname{Fiber~of}~ y ~\operatorname{under}~ f ~=~ f^{-1} (y) ~=~ \{ x \in X : f(x) = y \}.</math>
|}
In the special case where <math>f\!</math> is the indicator function <math>f_Q\!</math> of a set <math>Q \subseteq X,</math> the fiber of <math>\underline{1}</math> under <math>f_Q\!</math> is just the set <math>Q\!</math> back again:
{| align="center" cellpadding="8" width="90%"
| <math>\operatorname{Fiber~of}~ \underline{1} ~\operatorname{under}~ f_Q ~=~ f_Q ^{-1} (\underline{1}) ~=~ \{ x \in X : f_Q (x) = \underline{1} \} ~=~ Q.</math>
|}
|}
<pre>
<pre>
Fiber of %1% under f_Q = (f_Q)^(-1)(%1%) = {x in X : f_Q (x) = %1%} = Q.
Fiber of %1% under f_Q = (f_Q)^(-1)(%1%) = {x in X : f_Q (x) = %1%} = Q.