A correspondence, also called a one-to-one correspondence or a bijection, is a function from a set X to a set Y that has the property that for any element y in Y there is a unique element x in X such that f(x) = y. Most mathematical definitions of equivalence are correspondences that preserve some structure. For example, two groups are equivalent if there exists a correspondence between them that preserves their group structures. If there exists a correspondence between two sets, then they are said to have the same cardinality. A set X is said to be infinite if there is a subset Y of X such that Y is not equal to X, Y is not empty, and there is a correspondence between X and Y. The idea of correspondence originated with Bernhard Bolzano in the 1850s and was used extensively by Georg Cantor in his study of cardinality, infinite sets, and ordinal numbers. Cantor revolutionized the mathematical concept of infinity, but his contemporaries were hostile to his ideas and failed to realize their significance. Nowadays, Cantor's arguments are used without hesitation, and correspondences are used frequently in every area of mathematics.