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:
“I make this direct statement to the American people that there is far less chance of the United States getting into war, if we do all we can now to support the nations defending themselves against attack by the Axis than if we acquiesce in their defeat, submit tamely to an Axis victory, and wait our turn to be the object of attack in another war later on.”
—Franklin D. Roosevelt (1882–1945)
“To [secure] to each labourer the whole product of his labour, or as nearly as possible, is a most worthy object of any good government.”
—Abraham Lincoln (1809–1865)
“Children, who play life, discern its true law and relations more clearly than men, who fail to live it worthily, but who think that they are wiser by experience, that is, by failure.”
—Henry David Thoreau (1817–1862)