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Deﬁne the relation R+ by a R+ b if and only if there is a trip from a to b. Then clearly a R+ b if and only if there is some path from a to b in the directed graph. For instance, there is a path from Manchester to Rome, but no path from Rome to Manchester. We would like to calculate R + from R. Such a relation is called the transitive closure of R, since it is clearly transitive, and is in fact a special relation in the sense that it is the smallest transitive relation containing R. We can express the relation R + in terms of R using relational composition: a R+ b if and only if there is a path of length n from a to b, for some n ≥ 1. Another way of making this statement is to deﬁne the interim relations Rn : a Rn b if and only if there is a path of length n from a to b. Another way of deﬁning Rn is R1 = R R2 = R ◦ R R3 = R ◦ R2 = R2 ◦ R, since ◦ is associative . . . Rn = R ◦ Rn−1 = R ◦ . . . ◦ R, n times . . . Therefore, we have a R+ b if and only if ∃n ≥ 1. a Rn b: moreover R+ = R ∪ R2 ∪ . . . ∪ R n ∪ . . . = Rn
n≥1 There are many other natural examples of transitive closure, such as the following examples: 28 ...
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This note was uploaded on 01/02/2010 for the course MATH Math2009 taught by Professor Koskesh during the Spring '09 term at SUNY Empire State.
 Spring '09
 Koskesh
 Math

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