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:
“A concern with parenting...must direct attention beyond behavior. This is because parenting is not simply a set of behaviors, but participation in an interpersonal, diffuse, affective relationship. Parenting is an eminently psychological role in a way that many other roles and activities are not.”
—Nancy Chodorow (20th century)
“Poetry is the only life got, the only work done, the only pure product and free labor of man, performed only when he has put all the world under his feet, and conquered the last of his foes.”
—Henry David Thoreau (1817–1862)
“Words are but symbols for the relations of things to one another and to us; nowhere do they touch upon absolute truth.”
—Friedrich Nietzsche (1844–1900)