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:
“Art need no longer be an account of past sensations. It can become the direct organization of more highly evolved sensations. It is a question of producing ourselves, not things that enslave us.”
—Guy Debord (b. 1931)
“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)
“In the relations of a weak Government and a rebellious people there comes a time when every act of the authorities exasperates the masses, and every refusal to act excites their contempt.”
—John Reed (1887–1920)