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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, the transitive closure of a binary relation R on a set X is the smallest transitive relation on X that contains R (Lidl and Pilz 1998:337). If the original relation is transitive, the transitive closure will be that same relation; otherwise, the transitive closure will be 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."