IV. Product of Sets

Reference: Topology and Modern Analysis — G. F. Simmons, Tata McGraw Hill, India.

We shall often have occasion to weld together the sets of a given class into a single new set called their product (or their Cartesian product). The ancestor of this concept is the coordinate plane of analytic geometry, that is, a plane equipped with the normal rectangular coordinate axes. We give a brief description of this fundamental idea with a view to paving the way for our discussion of product of sets in general.

First, a few preliminary comments about the real line. We have already used this term several times before without any explanations, and of course what we mean by it is an ordinary geometric line whose points have been identified with — or coordinatized by — the set R of all real numbers. We use the letter R to denote the real line as well as the set of all real numbers, and we often speak of real numbers as if they were points on the real line, and of points on the real line as if they were real numbers. Let no be deceived into thinking that the real line is a simple thing, for its structure is exceedingly intricate. Our present view of it, however, is as naive and uncomplicated as the picture of it. Generally speaking, we assume that the reader is familiar with the simpler properties of the real line — those relating to inequalities (see problems section) and the basic algebraic operations of addition, subtraction, multiplication and division. One of the most significant facts about the real number system is perhaps less well known. This is the so-called least upper bound property. It asserts that every non empty set of real numbers which has an upper bound has a least upper bound. It is an easy consequence of this fact that a non empty set of real numbers which has a lower bound has a greatest lower bound. All these matters are developed rigorously on the basis of a small number of axioms, and detailed treatments can often be found in books on elementary abstract algebra.

To construct the coordinate plane, we now proceed as follows. We take two identical replicas of the real line, which we call the x axis and the y axis, and paste them on a plane at right angles to one another in such a way that they cross at the zero point on each. We know that usual picture. Now, let P be a point in the plane. We project P perpendicularly onto points Px and Py on the axes. If x and y are the coordinates of Px and Py on their respective axes, this process leads us from the point P to the uniquely determined ordered pair (x,y) of real numbers, where x and y are called the x coordinate and y coordinate of the point P. We can reverse the process, and starting with the ordered pair of real numbers, we can recapture the point. This is the manner in which we establish the familiar one-to-one correspondence between points P in the plane and ordered pairs (x,y) of real numbers. In fact, we think of a point in the plane (which is a geometric object) and its corresponding ordered pair of real numbers (which is an algebraic object) as being — to all intents and purposes — identical with one another. The essence of analytic geometry lies in the possibility of this exploiting this identification by using algebraic tools in geometric arguments and giving geometric interpretations to algebraic calculations.

The conventional attidute towards the coordinate plane in analytic geometry is that the geometry is the focus of interest and the algebra of ordered pairs is only a convenient tool. Here, we reverse this point of view. For us, the coordinate plane is defined to be the set of all ordered pairs (x,y) of real number. We can satisfy our desire for visual images by using the usual picture of the plane and by calling such an ordered pair a point, but this geometric lnaguage is more a convenience than a necessity.

Our notation for the coordinate plane is R \times R or R^{2}. This symbolism reflects the idea that the coordinate plane is the result of multiplying together two replicas of the real line R.

It is perhaps necessary to comment on one possible source of misunderstanding. When we speak of R^{2} as a plane, we do so only to establish an intuitive bond with the reader’s previous experience in analytic geometry. Our present attitude is that R^{2} is a pure set and has no structure whatsoever, because no structure has as yet been assigned to it. We remarked earlier (with deliberate vagueness) that a space is a set to which has been added some kind of algebraic or geometric structure. Later, we shall convert the set R^{2} into the space of analytic geometry by defining the distance between any two points (x_{1}, y_{1}) and (w_{2}, y_{2}) to be

\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}

This notion of distance endows the set R^{2} with a certain “spatial” character, which we shall recognize by calling the resulting space the Euclidean plane instead of the coordinate plane.

We assume that the reader is fully acquainted with the way in which the set C of all complex numbers can be identified (as a set) with the coordinate plane R^{2}. If z is a complex number, and if z has the standard form x+iy where x and y are real numbers, then we identify z with the ordered pair (x,y) and thus, with an element of R^{2}. The complex numbers, however, are much more than merely a set. They constitute a number system with the operations of addition, multiplication, conjugation, etc. When the coordinate plane R^{2} is thought of as consisting of complex numbers and is enriched by the algebraic structure it acquires in this way, it is called the complex plane. The letter C is used to denote either the set C of all complex numbers or the complex plane. Later, we shall make a space out of the complex plane.

Suppose now that X_{1} and X_{2} are two non empty sets. By analogy with our above discussion, their product X_{1} \times X_{2} is defined by the set of all ordered pairs (x_{1}, x_{2}) where x_{1} is in X_{1} and x_{2} is in X_{2}. In spite of the arbitrary nature of X_{1} and X_{2}, their product can be represented by a picture similar to the XY plane. The term product is applied to this set, and it is thought of as a result of multiplying together the sets X_{1} and X_{2} for the following reason: if X_{1} and X_{2} are finite sets with m and n elements, then clearly X_{1} \times X_{2} has mn elements. If f: X_{1} \rightarrow X_{2} is a mapping with domain X_{1} and range X_{2}, its graph is a subset of X_{1} \times X_{2} which consists of all ordered pairs of the form (x_{1}, f(x_{1})). We observe that this is an appropriate generalization of the concept of a graph of a function as it occurs in elementary mathematics.

This definition of the product of two sets extends easily to the definiion of product of n sets for any positive integer n. If X_{1}, X_{2}, X_{3}, \ldots, X_{n} are n sets where n is any positive integer, then their product X_{1} \times X_{2} \times X_{3} \times \ldots \times X_{n} is the set of all ordered tuples (x_{1}, x_{2}, \ldots, x_{n}) where x_{i} is in the set X_{i} for each subscript i. If the X_{i}‘s are all replicas of a single set X, that is, if

X_{1} = X_{2} = \ldots = X_{n} = X,

then their product is usually denoted by the symbol X^{n}.

These ideas specialize directly to yield the important sets R^{n} and C^{n}. R is just R, the real line; and R^{2} is the coordinate plane; R^{3} is the set of all ordered triples of real numbers — the set which underlies solid analytic geometry, and we assume that the reader is familiar with the manner in which this set arises, through the introduction of a rectangular coordinate system into three dimensional space. We can draw pictures here, just as in the case of the coordinate plane and use geometric language here as much as we please, but it must be understood that the mathematics of this set is the mathematics of ordered set of triples of real numbers and that pictures are merely an aid to the intuition. Once we fully this grasp this point of view, there is no difficulty whatsoever in advancing at once to the study of the set R^{n}, study of n-tuples of real numbers for any positive integer n. It is quite true that when n is greater than 3, it is no longer possible to draw the same kind of intuitively rich pictures, but at worst this is merely an inconvenience. We can and do continue to use suggestive geometric language, so all is not lost. The set C^{n} is defined similarly; it is the set of all ordered n-tuples (z_{1}, z_{2}, \ldots, z_{n}) of complex numbers. Both R^{n} and C^{n} are of fundamental importance in analysis and topology.

We emphasized that for the present the coordinate plane is to be considered merely as a set, but not as a space. Similar remarks apply to R^{n} and C^{n}. In due conrse, we shall impart form and content to each of these sets by suitable definitions. We shall convert them into Euclidean and n-unitary space which underlie and motivate so many developments in pure mathematics, and we shall explore some aspects of their algebraic and topological structures, But, as of now — and this is the point of view we insist on — they do not have any structure at all; they are merely sets.

As the reader doubtless suspects, we need not consider only products of finite classes of sets. The needs of topology compel us to extend these ideas to arbitrary classes of sets.

We defined the product X_{1} \times X_{2} \times \ldots \times X_{n} to be the set of all ordered n-tuples (x_{1}, x_{2}, \ldots, x_{n}) such that x_{i} is in X_{i} for each subscript i. To see how to extend this definition, we reformulate it as follows: We have an nidex set I, consisting of the integers from 1 to n, and corresponding to each index (or subscript) i we have a non-empty set X_{i}. The n-tuple (x_{1}, x_{2}, x_{3}, \ldots, x_{n}) is simply a function (call it x) defined on the index set I, with the restriction that its value x(i)=x_{i} is an element of the set X_{i} for each i in I. Our point of view here is that the function x is completely determined by and is essentially equivalent to the array (x_{1}, x_{2}, x_{3}, \ldots, x_{n}) of its values.

The way is now open for the definition of products in their full generality. Let \{ X_{i}\} be a non-empty class of non-empty sets, indexed by the elements i of an index set I. The sets X_{i} need not be different from one another; indeed, it may happen that they are all identical replicas of a single set, distinguished only different indices. The product of the sets X_{i}, written P_{i \in I}X_{i} is defined to be the set of all functions x defined on I such that x(i) is an element of the set X_{i} for each index i. We call X_{i} the ith coordinate set. When there can be no misunderstanding about the index set, the symbol P_{i \in I}X_{i} is often abbreviated to P_{i}X_{i}. The definition we have just given requires that each coordinate set be non-empty before the product can be formed. It will be useful if we extend this definition slightly by agreeing that if any of the X_{i}‘s are empty, then P_{i}X_{i} is also empty.

This approach to the idea of a product of a class of sets, by means of functions defined on the index set is useful mainly in giving the definition. In practice, it is much more convenient to use the subscript notation x_{i} instead of the function notation x(i). We then interpret the product P_{i}X_{i} as made up of elements x, each of which is specified by the exhibited array \{ x_{i}\} of it values in the respective coordinate sets X_{i}. We call x_{i} the ith coordinate of the element x = \{ x_{i}\}.

The mapping p_{i} of the product P_{i}X_{i} onto its ith coordinate set X_{i} which is defined by p_{i}(x) = x_{i} — that is, the mapping whose value at an arbitrary element of the product is the ith coordinate of that element — is called the projection onto the ith coordinate set. The projection p_{i} selects the ith coordinate of each element in its domain. There is clearly one projection for each element of the index set I, and the set of all projections plays an important role in the general theory of topological spaces.

We will continue with exercises on this topic in a later blog.

Regards,

Nalin Pithwa

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