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Both sides previous revision Previous revision Next revision | Previous revision Next revision Both sides next revision | ||
sav08:sets_and_relations [2009/03/10 14:13] vkuncak |
sav08:sets_and_relations [2009/03/10 20:29] vkuncak |
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S\bullet r = \{ y.\ \exists x. x \in S \land (x,y) \in r \} | S\bullet r = \{ y.\ \exists x. x \in S \land (x,y) \in r \} | ||
\] | \] | ||
+ | |||
==== Transitive Closure ==== | ==== Transitive Closure ==== | ||
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or, equivalently, the least relation $s$ (with respect to $\subseteq$) that satisfies | or, equivalently, the least relation $s$ (with respect to $\subseteq$) that satisfies | ||
\[ | \[ | ||
- | \Delta_A\ \cup\ (s \circ s)\ \subseteq\ s | + | \Delta_A\ \cup\ r \cup (s \circ s)\ \subseteq\ s |
\] | \] | ||
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- | ==== More Properties ==== | + | |
+ | ==== Range, Image, and Composition ==== | ||
\[ | \[ |