Existence and Description
For any relation R, the transitive closure of R always exists. To see this, note that the intersection of any family of transitive relations is again transitive. Furthermore, there exists at least one transitive relation containing R, namely the trivial one: X × X. The transitive closure of R is then given by the intersection of all transitive relations containing R.
We can describe the transitive closure of R in more concrete terms as follows, intuitively constructing it step by step. To start, define
and, for ,
Each set in this construction takes the relation from the previous set, and adds any new elements necessary to make chains that are "one step" more transitive. The relation R+ is made by taking all of the together, which we write as
To show that R+ is the transitive closure of R, we must show that it contains R, that it is transitive, and that it is the smallest set with both of those characteristics.
- : contains all of the, so in particular contains .
- is transitive: every element of is in one of the, so must be transitive by the following reasoning: if and, then from composition's associativity, (and thus in ) because of the definition of .
- is minimal: Let be any transitive relation containing, we want to show that . It is sufficient to show that for every, . Well, since contains, . And since is transitive, whenever, according to the construction of and what it means to be transitive. Therefore, by induction, contains every, and thus also .
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