# 28 - Manchester Knock London Dubin Paris Edinburgh Rome...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Manchester Knock London Dubin Paris Edinburgh Rome Madrid 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 ...
View Full Document

## This note was uploaded on 03/15/2010 for the course MATH 2009 taught by Professor Kooskash during the Spring '10 term at Sul Ross.

Ask a homework question - tutors are online