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)
“Man’s main task in life is to give birth to himself, to become what he potentially is. The most important product of his effort is his own personality.”
—Erich Fromm (1900–1980)
“So soon did we, wayfarers, begin to learn that man’s life is rounded with the same few facts, the same simple relations everywhere, and it is vain to travel to find it new.”
—Henry David Thoreau (1817–1862)