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:
“...I have never known a “movement” in the theater that did not work direct and serious harm. Indeed, I have sometimes felt that the very people associated with various “uplifting” activities in the theater are people who are astoundingly lacking in idealism.”
—Minnie Maddern Fiske (1865–1932)
“Labor is work that leaves no trace behind it when it is finished, or if it does, as in the case of the tilled field, this product of human activity requires still more labor, incessant, tireless labor, to maintain its identity as a “work” of man.”
—Mary McCarthy (1912–1989)
“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)