Note: the following has been abstracted from the Grolier Encyclopedia.

Set Theory

Any collection of objects is called a set, and set theory is the study of the relationships existing among sets. Set theory underlies the language and concepts of modern mathematics--both pure and applied. The study of sets, especially infinite ones, has also become a fascinating branch of mathematics in its own right. Set theory began with the work of Georg Cantor in the 19th century, but its roots in logic go back much further--to Aristotle and Plato.

The prevailing view in mathematics today is that every mathematical object can ultimately be described as some sort of set. A set may be specified in one of two basic ways. The roster method, or tabulation method, simply lists all the elements in the set. The descriptive method, or set-builder notation, gives a rule for determining which things are in the desired set and which are not. The set is then designated by a pair of braces (curly brackets) surrounding its description. Bear in mind that not every description that seems to make sense actually denotes a set. If it did, many inconsistencies, such as Russell's Paradox, would arise. Axiomatic set theory studies the fine-tuning that must be made in the definitions and axioms for sets in order to prevent such inconsistencies. Different systems of axioms are possible. In general, such systems deny the existence of sets that are so large that they contain themselves.

Sets can be named, usually with letters. The set containing no elements, called the Null Set or empty set, is denoted by the Greek lower-case letter phi. Two sets are equal if and only if they contain exactly the same elements. If every element of a set A is also an element of the set B, set A is a subset of B. The intersection of A and B is the set of all elements that are in both A and B. The union of A and B is the set of all elements that are either in A, or in B, or in both. The complement of B in A is A - B, the set of all elements in A that are not in B. The power set of A is the set of all subsets of A. It always contains the empty set and the whole original set A.

If A and B are two sets, their Cartesian product A X B is the set of all ordered pairs (a, b) where a is an element of A and b is an element of B. Note that (a, b) = (c, d) if and only if a = c and b = d. A relationship R between A and B is simply a subset of A X B. If (a, b) is an element of R, we write aRb. A function (also called a mapping) from A to B is simply a relation R with the property that for each element a in A there is one and only one element (x, y) in R such that x = a. A function may be thought of as a rule that associates a unique element of B to every element a in A. If R is a function, we write R(a) = b instead of aRb.

In addition to its applications to logic, computer science, and other branches of mathematics, set theory is of great value because of the clarity it brings to investigations concerning infinity. Using the concept of a function, we can compare the sizes of different sets. Let f be a function from A to B. The function f is said to be injective if for any element b in B there is at most one element a in A such that f (a) = b. We say f is bijective (giving a one-to-one correspondence) if for each element b in B there is exactly one element a in A such that f (a) = b. With this nomenclature we can now define the cardinality of a set: A and B have the same cardinality if and only if there exists a bijective function from A to B. If there is an injective function from A to B, we say that the cardinality of A is less than or equal to the cardinality of B. Thus the cardinality of a set is the measure of the size of the set. For instance, all sets with exactly 13 elements have the same cardinality. Not all infinite sets, however, have the same cardinality. Cantor's great discovery was that the cardinality of the set of real numbers is greater than the cardinality of the set of integers.