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:
“Bryan is the least of a liar I know in public life. I have always found him direct and honest, and he never goes back on what he has said to me in private—a rare thing, if found, in public men. I found him purely frank.”
—William Howard Taft (1857–1930)
“A gentleman opposed to their enfranchisement once said to me, “Women have never produced anything of any value to the world.” I told him the chief product of the women had been the men, and left it to him to decide whether the product was of any value.”
—Anna Howard Shaw (1847–1919)
“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)