Bernhard Bolzano defined cardinality in his book Paradoxes of the Infinite (1851). The cardinality of a set is, roughly speaking, its size. Precisely, the cardinality of X is said to be less than or equal to that of Y if there is a function f from X to Y with the property that if x and x' are elements of X with f(x) = f(x') then x = x'. Such a function is said to be one to one or injective. Equivalently, the cardinality of X is said to be less than or equal to that of Y if there is a function f from Y to X with the property that if x is an element of X then there is an element y of Y with f(y) = x. Such a function is said to be onto or surjective. If the cardinality of X is less than...