Transitive closure

In mathematics, the transitive closure of a binary relation on a set X is the smallest transitive relation on X that contains .

In more concrete terms the transitive closure of R is the relation R* such that xR*y if xRy, or if xRz for some z with zRy, or if xRz and zRw and wRy for some z and w in X, and so on for any number of intermediates. If X is the set of humans (alive or dead) and R is the relation 'parent of', then xR*y means y is a direct descendant of x.

Examples

• If xRy means x is the parent of y, then the transitive closure of R is the relation "x is an ancestor of y."

• If xRy means "there is a regular direct airplane flight from airport x to airport y", then the transitive closure of R is the relation "it is possible to fly from x to y in one or more flights."