function is called injective (or one-to-one) if it maps any two different elements of to different elements of , surjective (or onto) if for every element of , there is at least one element of that maps to it, and bijective (or a one-to-one correspondence) if the function is both injective and surjective — in this case, each element of is paired with a unique element of , and each element of is paired with a unique element of , so that there are no unpaired elements. An injective function is called an injection, a surjective function is called a surjection, and a bijective function is called a bijection or one-to-one correspondence. Cardinality The cardinality of a set , denoted , is the number of members of . For example, if , then . Repeated members in roster notation are not counted, so , too. More formally, two sets share the same cardinality if there exists a one-to-one correspondence between them. The cardinality of the empty set is zero. Infinite sets and infinite cardinality The list of elements of some sets is endless, or infinite. For example, the set of natural numbers is infinite. In fact, all the special sets of numbers mentioned in the section above are infinite. Infinite sets have infinite cardinality. Some infinite cardinalities are greater than others. Arguably one of the most significant results from set theory is that the set of real numbers has greater cardinality than the set of natural numbers. Sets with cardinality less than or equal to that of are called countable sets; these are either finite sets or countably infinite sets (sets of the same cardinality as ); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than that of are called uncountable sets. However, it can be shown that the cardinality of a straight line (i.e., the number of points on a line) is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space. The continuum hypothesis The continuum hypothesis, formulated by Georg Cantor in 1878, is the statement that there is no set with cardinality strictly between the cardinality of the natural numbers and the cardinality of a straight line. In 1963, Paul Cohen proved that the continuum hypothesis is independent of the axiom system ZFC consisting of Zermelo–Fraenkel set theory with the