Set Theory

I

Introduction

Set Theory, branch of mathematics concerned with the abstract properties of sets, or collections of objects. A set can be a physical grouping, such as the set of all people present in a room; or a conceptual aggregate, such as the set of all British prime ministers, past and present. Each of these sets is defined by a property that its members share, but it is possible for a set to be a completely arbitrary collection. For the purpose of studying the mathematical properties of sets, it does not matter whether their members are concrete or abstract, real or fictitious, or whether these different types of things are mixed together: for example, a set can be defined simply as consisting of the even numbers, a platypus, and Robinson Crusoe. Difficulties arise at the frontiers of the subject, however, in connection with the question of whether sets can be members of themselves.

Set theory was first given formal treatment by the German mathematician Georg Cantor in the 19th century. The set concept is one of the most basic in mathematics, even more primitive than the process of counting, and is found, explicitly or implicitly, in every area of pure and applied mathematics. Explicitly, the principles and terminology of sets are used to make mathematical statements more clear and precise and to clarify concepts such as the finite and the infinite.

II

Definitions

A set is an aggregate, class, or collection of objects, which are called the elements of the set. In symbols, a e S means that the element a belongs to or is contained in the set S, or that the set S contains the element a. A set S is defined if, given any object a, one and only one of these statements holds: a e S or a  S (that is, a is not contained in S).

A set is frequently designated by the symbol S = { }, with the braces including the elements of S either by writing all of them in explicitly or by giving a formula, rule, or statement that describes all of them. Thus, S₁ = {2, 4}; S₂ = {2, 4, 6, ..., 2n, ...} = {all positive even integers}; S₃ = {x | x² - 6x + 11 ≥ 3}; S₄ = {all living males named John}. S₃ is read as the set of all xs such that x² - 6x + 11 ≥ 3.

A

Subset and Superset

If every element of a set R also belongs to a set S,R is a subset of S, and S is a superset of R; in symbols, R Í S, or S Ê R. A set is thus both a subset and a superset of itself. If R Í S, but at least one element in S is not in R, then R is called a proper subset of S, and S is a proper superset of R; in symbols, R Ì S, S ÉR. If R Í S and S Ê R, that is, if every element of one set is an element of the other, then R and S are the same, and one writes R = S. Thus, in the examples cited above, S₁ is a proper subset of S₂.

B

Union and Intersection

If A and B are two subsets of a set S, the elements found in A or in B or in both form a subset of S called the union of A and B, written AÈ B. The elements common to A and B form a subset of S called the intersection of A and B, written A Ç B. If A and B have no elements in common, the intersection is empty; it is convenient, however, to think of the intersection as a set, designated by Æ and called the empty, or null, set. Thus, if A = {2, 4, 6}, B ={4, 6, 8, 10}, and C = {10, 14, 16, 26}, then A È B = {2, 4, 6, 8, 10}, A È C = {2, 4, 6, 10, 14, 16, 26}, A Ç B = {4, 6}, A Ç C = Æ.