Unformatted text preview: b R v . Now a R x , x R b and b R v , so a R v by transitivity. Therefore v ∈ [ a ] and so [ b ] ⊆ [ a ] . But [ a ] ⊆ [ b ] using a similar argument, so [ a ] = [ b ] . ± 3.7 Transitive Closure Consider the following situation. There are various ±ights between various cities. ²or any two cities, we wish to know whether it is possible to ±y from one to the other allowing for changes of plane. We can model this by deFning a set City of cities and a binary relation R such that a R b if and only if there is a direct ±ight from a to b . This relation may be represented as a directed graph with the cities as nodes, as in the following example: 27...
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 Spring '09
 Koskesh
 Math, Sets, Equivalence relation, Transitive relation, equivalence classes, ﬁve disjoint subsets

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