transitive closure

In mathematics, the transitive closure of a binary relation R on a set X is the transitive relation R⁺ on set X such that R⁺ contains R and R⁺ is minimal (Lidl and Pilz 1998:337). If the binary relation itself is transitive, then the transitive closure is that same binary relation; otherwise, the transitive closure is a different relation. For example, if X is a set of airports and x R y means "there is a direct flight from airport x to airport y", then the transitive closure of R on X is the relation R⁺: "it is possible to fly from x to y in one or more flights."