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:
“Of course it is of no use to direct our steps to the woods, if they do not carry us thither. I am alarmed when it happens that I have walked a mile into the woods bodily, without getting there in spirit.... What business have I in the woods, if I am thinking of something out of the woods?”
—Henry David Thoreau (1817–1862)
“A product of the untalented, sold by the unprincipled to the utterly bewildered.”
—Al Capp (1909–1979)
“If one could be friendly with women, what a pleasure—the relationship so secret and private compared with relations with men. Why not write about it truthfully?”
—Virginia Woolf (1882–1941)