In combinatorics, double counting, also called counting in two ways, is a combinatorial proof technique for showing that two expressions are equal by demonstrating that they are two ways of counting the size of one set. In this technique, which van Lint & Wilson (2001) call “one of the most important tools in combinatorics,” one describes a finite set X from two perspectives leading to two distinct expressions for the size of the set. Since both expressions equal the size of the same set, they equal each other.
Famous quotes containing the words double and/or counting:
“The worst feature of this double consciousness is, that the two lives, of the understanding and of the soul, which we lead, really show very little relation to each other; never meet and measure each other: one prevails now, all buzz and din; and the other prevails then, all infinitude and paradise; and, with the progress of life, the two discover no greater disposition to reconcile themselves.”
—Ralph Waldo Emerson (1803–1882)
“Is it not manifest that our academic institutions should have a wider scope; that they should not be timid and keep the ruts of the last generation, but that wise men thinking for themselves and heartily seeking the good of mankind, and counting the cost of innovation, should dare to arouse the young to a just and heroic life; that the moral nature should be addressed in the school-room, and children should be treated as the high-born candidates of truth and virtue?”
—Ralph Waldo Emerson (1803–1882)