The transitive closure of a binary relation 
on a set 
is the minimal transitive relation 
on 
that contains 
. Thus 
for any elements 
and 
of 
provided that there exist 
, 
, ..., 
with 
, 
, and 
for all 
.
The transitive closure 
of a graph is a graph
which contains a graph edge 
whenever there is a directed graph
path from 
to 
(Skiena 1990, p. 203). The transitive closure graph can be computed in the Wolfram Language using TransitiveClosureGraph[g].
In set theory, the transitive closure 
of a set 
is the smallest transitive set
containing 
as a subset. If 
denotes the union of the
elements of 
, then
 |
Thus 
and every element of an element
of 
is again in 
.
A convention that requires the transitive closure to contain 
as an element instead uses 
.
