Set Theory
Set theory is concerned with understanding those properties of sets that are independent of the particular elements that make up the sets. Thus the axioms and theorems of set theory apply to all sets in general, whether they are composed of numbers or physical objects. The foundations of set theory were largely developed by the German mathematician George Cantor in the latter part of the nineteenth century. The generality of set theory leads to few direct practical applications. Instead, precisely because of its generality, portions of the theory are used in developing the algebra of groups, rings, and fields, as well as, in developing a logical basis for calculus, geometry, and topology. These branches of mathematics are all applied extensively in the fields of physics, chemistry, biology, and electrical and computer engineering.
Definitions
A set is a collection. As with any collection, a set is composed of objects, called members or elements. The elements of a set may be physical objects or mathematical objects. A set may be composed of baseball cards, salt shakers, tropical fish, numbers, geometric shapes, or abstract mathematical constructs such as functions. Even ideas may be elements of a set. In fact, the elements of a set are not required to have anything in common except that they belong to the same set. The collection of all the junk at a rummage sale is a perfectly good set, but one in which few of the elements have anything in common, except that someone has gathered them up and put them in a rummage sale.
In order to specify a set and its elements as completely and unambiguously as possible, standard forms of notation (sometimes called set-builder notation) have been adopted by mathematicians. For brevity a set is usually named using an uppercase Roman letter, such a S. When defining the set S, curly brackets {} are used to enclose the contents, and the elements are specified, inside the brackets. When convenient, the elements are listed individually. For instance, suppose there are 5 items at a rummage sale. Then the set of items at the rummage sale might be specified by R={basketball, horseshoe, scooter, bow tie, hockey puck}. If the list of elements is long, the set may be specified by defining the condition that an object must satisfy in order to be considered an element of the set. For example, if the rummage sale has hundreds of items, then the set R may be specified by R = {I: I is an item in the rummage sale). In this notation, I corresponds to an element of the set. The definition is read "R equals the set of all I such that I is an item in the rummage sale." If the set has an infinite number of elements it is specified similarly, such as S = {x: x is a real number and 0 < x < 1}. This is the set of all x such that x is a real number, and 0 is less than x, and x is less than 1. The special symbol is given to the set with no elements, called the empty set or null set. Finally, it means that x is an element of the set A, and means that x is not an element of the set A.
Properties
Two sets S and T are equal, if every element of the set S is also an element of the set T, and if every element of the set T is also an element of the set S. This means that two sets are equal only if they both have exactly the same elements. A set T is called a proper subset of S if every element of T is contained in S, but not every element of S is in T. That is, the set T is a partial collection of the elements in S.
In set notation this is written T S and read "T is contained in S." S is sometimes referred to as the parent or universal set. Also, S is a subset of itself, called an improper subset. The complement of a subset T is that part of S that is not contained in T, and is written T'. Note that if T' is the empty set, then S and T are equal.
Sets are classified by size, according to the number of elements they contain. A set may be finite or infinite. A finite set has a whole number of elements, called the cardinal number of the set. Two sets with the same number of elements have the same cardinal number. To determine whether two sets, S and T, have the same number of elements, a one-to-one correspondence must exist between the elements of S and the elements of T. In order to associate a cardinal number with an infinite set, the transfinite numbers were developed. The first transfinite number 0, is the cardinal number of the set of integers, and of any set that can be placed in one-to-one correspondence with the integers. For example, it can be shown that a one-to-one correspondence exists between the set of rational numbers and the set of integers. Any set with cardinal number 0 is said to be a countable set. The second transfinite number 1 is the cardinal number of the real numbers. Any set in one-to-one correspondence with the real numbers has a cardinal number of 1, and is referred to as uncountable. The irrational numbers have cardinal number 1. Some interesting differences exist between subsets of finite sets and subsets of infinite sets. In particular, every proper subset of a finite set has a smaller cardinal number than its parent set. For example, the set S = {1,2,3,4,5,6,7,8,9,10} has a cardinal number of 10, but every proper subset of S (such as {1,2,3,4}) has fewer elements than S and so has a smaller cardinality. In the case of infinite sets, however, this is not true. For instance, the set of all odd integers is a proper subset of the set of all integers, but it can be shown that a one-to-one correspondence exists between these two sets, so that they each have the same cardinality.
A set is said to be ordered if a relation (symbolized by <) between its elements can be defined, such that for any two elements of the set:
1) either b < c or c < b for any two elements
2) b < b has no meaning
3) if b < c and c < d then b < d.
In other words, an ordering relation is a rule by which the members of a set can be sorted. Examples of ordered sets are: the set of positive integers, where the symbol (<) is taken to mean less than; or the set of entries in an encyclopedia, where the symbol (<) means alphabetical ordering; or the set of U.S. World Cup soccer players, where the symbol (<) is taken to mean shorter than. In this last example the symbol (<) could also mean faster than, or scored more goals than, so that for some sets more than one ordering relation can be defined.
Operations
In addition to the general properties of sets, there are three important set operations, they are union, intersection, and difference. The union of two sets S and T, written ST, is defined as the collection of those elements that belong to either S or T or both. The union of two sets corresponds to their sum.
The intersection of the sets S and T is defined as the collection of elements that belong to both S and T, and is written ST. The intersection of two sets corresponds to the set of elements they have in common, or in some sense to their product.
The difference between two sets, written S-T, is the set of elements that are contained in S but not contained in T.
If S is a subset of T, then S-T = , and if the intersection of S and T (ST) is the null set, then S-T = S.
Applications of set theory
Because of its very general or abstract nature, set theory has many applications in other branches of mathematics. In the branch called analysis, of which differential and integral calculus are important parts, an understanding of limit points and what is meant by the continuity of a function are based on set theory. The algebraic treatment of set operations leads to Boolean Algebra, in which the operations of intersection, union, and difference are interpreted as corresponding to the logical operations "and," "or," and "not," respectively. Boolean Algebra in turn is used extensively in the design of digital electronic circuitry, such as that found in calculators and personal computers. Set theory provides the basis of topology, the study of sets together with the properties of various collections of subsets.
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