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:
“England and France, Spain and Portugal, Gold Coast and Slave Coast, all front on this private sea; but no bark from them has ventured out of sight of land, though it is without doubt the direct way to India.”
—Henry David Thoreau (1817–1862)
“Culture is a sham if it is only a sort of Gothic front put on an iron building—like Tower Bridge—or a classical front put on a steel frame—like the Daily Telegraph building in Fleet Street. Culture, if it is to be a real thing and a holy thing, must be the product of what we actually do for a living—not something added, like sugar on a pill.”
—Eric Gill (1882–1940)
“She has problems with separation; he has trouble with unity—problems that make themselves felt in our relationships with our children just as they do in our relations with each other. She pulls for connection; he pushes for separateness. She tends to feel shut out; he tends to feel overwhelmed and intruded upon. It’s one of the reasons why she turns so eagerly to children—especially when they’re very young.”
—Lillian Breslow Rubin (20th century)