Direct Product of Binary Relations
On the Cartesian product of two sets with binary relations R and S, define (a, b) T (c, d) as a R c and b S d. If R and S are both reflexive, irreflexive, transitive, symmetric, or antisymmetric, relation T has the same property. Combining properties it follows that this also applies for being a preorder and being an equivalence relation. However, if R and S are total relations, T is in general not.
Read more about this topic: Direct Product
Famous quotes containing the words direct, product and/or relations:
“An inexperienced heraldist resembles a medieval traveler who brings back from the East the faunal fantasies influenced by the domestic bestiary he possessed all along rather than by the results of direct zoological exploration.”
—Vladimir Nabokov (1899–1977)
“Poetry, whose material is language, is perhaps the most human and least worldly of the arts, the one in which the end product remains closest to the thought that inspired it.... Of all things of thought, poetry is the closest to thought, and a poem is less a thing than any other work of art ...”
—Hannah Arendt (1906–1975)
“I know all those people. I have friendly, social, and criminal relations with the whole lot of them.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)