London
Edinburgh
Manchester
Dubin
Knock
Paris
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 composi
tion:
a R
+
b
if and only if there is a path of length
n
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 Spring '09
 Koskesh
 Math, SEPTA Regional Rail, Transitive relation, Jaguar Racing, Transitive closure, smallest transitive relation

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