Advanced Calculus with Applications in Statistics

John Wiley & Sons

Chapter One

The origin of the modern theory of sets can be traced back to the Russian-born German mathematician Georg Cantor (1845-1918). This chapter introduces the basic elements of this theory.

1.1. THE CONCEPT OF A SET

A set is any collection of well-defined and distinguishable objects. These objects are called the elements, or members, of the set and are denoted by lowercase letters. Thus a set can be perceived as a collection of elements united into a single entity. Georg Cantor stressed this in the following words: "A set is a multitude conceived of by us as a one."

If x is an element of a set A, then this fact is denoted by writing x [element of] A. If, however, x is not an element of A, then we write x [??] A. Curly brackets are usually used to describe the contents of a set. For example, if a set A consists of the elements [x.sub.1], [x.sub.2],..., [x.sub.n], then it can be represented as A = {[x.sub.1], [x.sub.2],..., [x.sub.n]}. In the event membership in a set is determined by the the satisfaction of a certain property or a relationship, then the description of the same can be given within the curly brackets. For example, if A consists of all real numbers x such that [chi square] > 1, then it can be expressed as A = {x|[chi square] > 1}, where the bar | is used simply to mean "such that." The definition of sets in this manner is based on the axiom of abstraction, which states that given any property, there exists a set whose elements are just those entities having that property.

Definition 1.1.1. The set that contains no elements is called the empty set and is denoted by [empty set].

Definition 1.1.2. A set A is a subset of another set B, written symbolically as A [subset] B, if every element of A is an element of B. If B contains at least one element that is not in A, then A is said to be a proper subset of B.

Definition 1.1.3. A set A and a set B are equal if A [subset] B and B [subset] A. Thus, every element of A is an element of B and vice versa.

Definition 1.1.4. The set that contains all sets under consideration in a certain study is called the universal set and is denoted by [Omega].

1.2. SET OPERATIONS

There are two basic operations for sets that produce new sets from existing ones. They are the operations of union and intersection.

Definition 1.2.1. The union of two sets A and B, denoted by A [union] B, is the set of elements that belong to either A or B, that is,

A [union] B = {x|x [element of] A or x [element of] B}.

This definition can be extended to more than two sets. For example, if [A.sub.1], [A.sub.2],..., [A.sub.n] are n given sets, then their union, denoted by [[union].sup.n.sub.i=1][A.sub.i], is a set such that x is an element of it if and only if x belongs to at least one of the [A.sub.i] (i = 1, 2,..., n).

Definition 1.2.2. The intersection of two sets A and B, denoted by A [intersection] B, is the set of elements that belong to both A and B. Thus

A [intersection] B = {x|x [element of] A and x [element of] B}.

This definition can also be extended to more than two sets. As before, if [A.sub.1], [A.sub.2],..., [A.sub.n] are n given sets, then their intersection, denoted by [[intersection].sup.n.sub.i]=1 [A.sub.i], is the set consisting of all elements that belong to all the [A.sub.i] (i = 1, 2,..., n).

Definition 1.2.3. Two sets A and B are disjoint if their intersection is the empty set, that is, A [intersection] B = [empty set].

Definition 1.2.4. The complement of a set A, denoted by [bar.A], is the set consisting of all elements in the universal set that do not belong to A. In other words, x [element of] [bar.A] if and only if x [??] A.

The complement of A with respect to a set B is the set B - A which consists of the elements of B that do not belong to A. This complement is called the relative complement of A with respect to B.

From Definitions 1.1.1-1.1.4 and 1.2.1-1.2.4, the following results can be concluded:

Result 1.2.1. The empty set [empty set] is a subset of every set. To show this, suppose that A is any set. If it is false that [empty set] [subset] A, then there must be an element in [empty set] which is not in A. But this is not possible, since [empty set] is empty. It is therefore true that [empty set] [subset] A.

Result 1.2.2. The empty set [empty set] is unique. To prove this, suppose that [[empty set].sub.1] and [[empty set].sub.2] are two empty sets. Then, by the previous result, [[empty set].sub.1] [subset] [[empty set].sub.2] and [[empty set].sub.2] [greater than or equal to] [[empty set].sub.2]. Hence, [[empty set].sub.1] = [[empty set].sub.2].

Result 1.2.3. The complement of [empty set] is [Omega]. Vice versa, the complement of [Omega] is [empty set].

Result 1.2.4. The complement of [bar.A] is A.

Result 1.2.5. For any set A, A [union] [bar.A] = [Omega] and A [intersection] [bar.A] = [empty set].

Result 1.2.6. A - B = A - A [intersection] B.

Result 1.2.7. A [union] (B [union] ITLITL) = (A [union] B) [union] ITLITL.

Result 1.2.8. A [intersection] (B [intersection] ITLITL) = (A [intersection] B) [intersection] ITLITL).

Result 1.2.9. A [union] (B [intersection] ITLITL) = (A [union] B) [intersection] (A [union] ITLITL).

Result 1.2.10. A [intersection] (B [union] ITLITL) = (A [intersection] B) [union] (A [intersection] ITLITL).

Result 1.2.11. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Result 1.2.12. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 1.2.5. Let A and B be two sets. Their Cartesian product, denoted by A x B, is the set of all ordered pairs (a, b) such that a [element of] A and b [element of] B, that is,

A x B = {(a, b)|a [element of] A and b [element of] B}.

The word "ordered" means that if a and c are elements in A and b and d are elements in B, then (a, b) = (c, d) if and only if a = c and b = d.

The preceding definition can be extended to more than two sets. For example, if [A.sub.1], [A.sub.2],..., [A.sub.n] are n given sets, then their Cartesian product is denoted by [x.sup.n.sub.i=1][A.sub.i] and defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Here, ([a.sub.1], [a.sub.2],..., [a.sub.n]), called an ordered n-tuple, represents a generalization of the ordered pair. In particular, if the [A.sub.i] are equal to A for i = 1, 2,..., n, then one writes [A.sup.n] for [x.sup.n.sub.i=1]A.

The following results can be easily verified:

Result 1.2.13. A x B = [empty set] if and only if A = [empty set] or B = [empty set].

Result 1.2.14. (A [union] B]) x ITLITL = (A x ITLITL) [union] (B x ITLITL).

Result 1.2.15. (A [intersection] B) x ITLITL = (A x ITLITL) [intersection] x (B x ITLITL).

Result 1.2.16. (A x B) [intersection] (ITLITL x D) = (A [intersection] ITLITL) x (B [intersection] D).

1.3. RELATIONS AND FUNCTIONS

Let A x B be the Cartesian product of two sets, A and B.

Definition 1.3.1. A relations [rho] from A to B is a subset of A x B, that is, [rho] consists of ordered pairs (a, b) such that a [element of] A and b [element of] B. In particular, if A = B, then [rho] is said to be a relation in A.

For example, if A = {7, 8, 9} and B = {7, 8, 9, 10}, then [rho] {(a, b)|a < b, a [element of] A, b [element of] B} is a relation from A to B that consists of the six ordered pairs (7, 8), (7, 9), (7, 10), (8, 9), (8, 10), and (9, 10).

Whenever [rho] is a relation and (x, y) [element of] [rho], then x and y are said to be [rho]-related. This is denoted by writing x p y.

Definition 1.3.2. A relation [rho] in a set A is an equivalence relation if the following properties are satisfied:

1. [rho] is reflexive, that is, a [rho] a for any a in A.

2. [rho] is symmetric, that is, if a [rho] b, then b [rho] a for any a, b in A.

3. [rho] is transitive, that is, if a [rho] b and b [rho] c, then a [rho] c for any a, b, c in A.

If [rho] is an equivalence relation in a set A, then for a given [a.sub.0] in A, the set

ITLITL([a.sub.0]) = {a [element of] A|[a.sub.0] [rho] a},

which consists of all elements of A that are [rho]-related to [a.sub.0], is called an equivalence class of [a.sub.0].

Result 1.3.1. a [element of] ITLITL(a) for any a in A. Thus each element of A is an element of an equivalence class.

Result 1.3.2. If ITLITL([a.sub.1]) and ITLITL([a.sub.2]) are two equivalence classes, then either ITLITL([a.sub.1]) = ITLITL([a.sub.2]), or ITLITL([a.sub.1) and ITLITL([a.sub.2]) are disjoint subsets.

It follows from Results 1.3.1 and 1.3.2 that if A is a nonempty set, the collection of distinct [rho]-equivalence classes of A forms a partition of A.

As an example of an equivalence relation, consider that a [rho] b if and only if a and b are integers such that a - b is divisible by a nonzero integer n. This is the relation of congruence modulo n in the set of integers and is written symbolically as a [equivalent to] b (mod n). Clearly, a [equivalent to] a (mod n), since a - a = 0 is divisible by n. Also, if a [equivalent to] b (mod n), then b [equivalent to] a (mod n), since if a - b is divisible by n, then so is b - a. Furthermore, if a [equivalent to] _ b (mod n) and b [equivalent to] c (mod n), then a [equivalent to] c (mod n). This is true because if a - b and b - c are both divisible by n, then so is (a - b) + (b - c) = a - c. Now, if [a.sub.0] is a given integer, then a [rho]-equivalence class of [a.sub.0] consists of all integers that can be written as a = [a.sub.0] + kn, where k is an integer. This in this example ITLITL([a.sub.0]) is the set {[a.sub.0] + kn|k [element of] J}, where J denotes the set of all integers.

Definition 1.3.3. Let [rho] be a relation from A to B. Suppose that [rho] has the property that for all x in A, if x [rho] y and x [rho] z, where y and z are elements in B, then y = z. Such a relation is called a function.

Thus a function is a relation [rho] such that any two elements in B that are [rho]-related to the same x in A must be identical. In other words, to each element x in A, there corresponds only one element y in B. We call y the value of the function at x and denote it by writing y = f(x). The set A is called the domain of the function f, and the set of all values of f(x) for x in A is called the range of f, or the image of A under f, and is denoted by f(A). In this case, we say that f is a function, or a mapping, from A into B. We express this fact by writing f: A [right arrow] B. Note that f(A) is a subset of B. In particular, if B = f(A), then f is said to be a function from A onto B. In this case, every element b in B has a corresponding element a in A such that b = f(a).

Definition 1.3.4. A function f defined on a set A is said to be a one-to-one function if whenever f([x.sub.1]) = f([x.sub.2]) for [x.sub.1], [x.sub.2] in A, one has [x.sub.1] = [x.sub.2]. Equivalently, f is a one-to-one function if whenever [x.sub.1] [not equal to] [x.sub.2], one has f([x.sub.1])[not equal to]f([x.sub.2]).

Continues...

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