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:
“Traditionally in American society, men have been trained for both competition and teamwork through sports, while women have been reared to merge their welfare with that of the family, with fewer opportunities for either independence or other team identifications, and fewer challenges to direct competition. In effect, women have been circumscribed within that unit where the benefit of one is most easily believed to be the benefit of all.”
—Mary Catherine Bateson (b. 1939)
“Everything that is beautiful and noble is the product of reason and calculation.”
—Charles Baudelaire (1821–1867)
“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)