Transitive Relations
Transitive relations are binary relations R on a single set X where for all a, b, c in X, aRb and bRc implies aRc. Transitive relations fall into two broad classes, equivalence relations and order relations. Equivalence relations are also symmetric and reflexive, while order relations are antisymmetric (complete order) or asymmetric (partial order) and may be reflexive (inclusive order) or anti-reflexive (strict order). The algebraic structure of equivalence relations builds on transformation groups; that of order relations builds on lattice theory. For more on relations and mathematics, from a philosophical standpoint, see Lucas (1999: chpt. 9).
Read more about this topic: Finitary Relation
Famous quotes containing the word relations:
“Major [William] McKinley visited me. He is on a stumping tour.... I criticized the bloody-shirt course of the canvass. It seems to me to be bad “politics,” and of no use.... It is a stale issue. An increasing number of people are interested in good relations with the South.... Two ways are open to succeed in the South: 1. A division of the white voters. 2. Education of the ignorant. Bloody-shirt utterances prevent division.”
—Rutherford Birchard Hayes (1822–1893)