Infinity (mathematics).

Infinity (mathematics). Infinity (mathematics), mathematical term derived from the theory of sets propounded by the German mathematician Georg Cantor. Sets may be divided into two classes, depending on whether the elements of a set can be put into a one-to-one correspondence (matched in a one-to-one way) with the elements of some proper subset. A set A is a proper subset of a set B if every element of A belongs to B but B has at least one element that does not belong to A. The elements of the set [1, 2, 3] cannot be matched in a one-to-one way with the elements of any one of its proper subsets; such a set is called a finite set. The elements of the set [2, 4, 6, ..., 2 n,...] can be matched in a one-to-one way with the elements of the proper subset [6, 8, 10, ..., 2n + 4, ...] by matching, for any positive integer n, the element 2n of the first set with the element 2n + 4 of the second. A set with this property is called an infinite set. Thus, the set N of all positive integers, the set R of all rational numbers, and the set Z of all real numbers are infinite sets. The elements of the sets N and R can be matched in a one-to-one way, so N and R have equal infinitudes of numbers; but N or R can be matched in a one-to-one way only with a proper subset of Z. Therefore, the infinitude of Z is "larger" than the infinitude of N. It can be shown that if S is any set, finite or infinite, the set T of subsets of S is a "larger" set; that is, the elements of S can be matched in a one-to-one way with a proper subset of T but not with T itself. The terms infinite and infinity are used in other, related ways. For example, in the infinite sequence 1, 4, 9, ..., of which the n th term, an, is equal to n 2, n = 1, 2, 3, ..., it is said that an becomes infinite as n becomes infinite, meaning that an is larger than any arbitrary preassigned number if n is larger than a certain value. In the infinite sequence 1, y, ? ..., of which the n th term, b n, is equal to 1/n, n = 1, 2, 3, ..., it is said that b n approaches 0 as n becomes infinite, meaning that the numerical , difference between b n and 0 can be made less than any arbitrary preassigned positive number if n is larger than a certain value. It also is said that f(x) = 1/(1 - x)2 approaches infinity, or becomes infinite, as x approaches 1 and that the function approaches 0 as x becomes infinite. Contributed By: James Singer Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.