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:
“One key, one solution to the mysteries of the human condition, one solution to the old knots of fate, freedom, and foreknowledge, exists, the propounding, namely, of the double consciousness. A man must ride alternately on the horses of his private and public nature, as the equestrians in the circus throw themselves nimbly from horse to horse, or plant one foot on the back of one, and the other foot on the back of the other.”
—Ralph Waldo Emerson (1803–1882)
“If all power is in the people, if there is no higher law than their will, and if by counting their votes, their will may be ascertained—then the people may entrust all their power to anyone, and the power of the pretender and the usurper is then legitimate. It is not to be challenged since it came originally from the sovereign people.”
—Walter Lippmann (1889–1974)