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)
“These facts have always suggested to man the sublime creed that the world is not the product of manifold power, but of one will, of one mind; and that one mind is everywhere active, in each ray of the star, in each wavelet of the pool; and whatever opposes that will is everywhere balked and baffled, because things are made so, and not otherwise.”
—Ralph Waldo Emerson (1803–1882)
“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)