Introduction to Algebraic Functions

The idea of correspondence is encountered regularly in everyday life. For example, to each book in a library there resembles the number of pages in the book. As another example, to each human being there resembles a birth date. To cite a third example, if the temperature of the air is recorded during a day, then at each instant of time there is a equivalent temperature.

The examples of correspondences we have given involve two sets X and Y. In our first example, X means the set of books in a library and Y the set of positive integers. For each book x in X there corresponds a positive integer y, namely the number of pages in the book. In the second example, if we let X denote the set of all human beings and Y the set of all possible dates, then to every person x in X there relates a birth date y.

We occasionally signify correspondences by drawings where the sets X and Y are signified by points within regions in a plane. The curved arrow designates that the element y of Y corresponds to the element x of X. We have pictured X and Y as different sets. However, X and Y may have elements in common. As a matter of fact, we often have X = Y.

Our examples show that to each x in X there corresponds one and only one y in Y; that is, y is exceptional for a given x. Though, the same element of Y may correspond to dissimilar elements of X. For example, two different books may have the same number of pages, two different people may have the same birthday, and so on.

In much of our work X and Y will be sets of real numbers. To demonstrate, let X and Y both signify the set R of real numbers, and to each real number x let us allocate its square x2. Thus to 3 we assign 9, to - 5 we assign 25, and so on. This gives us a correspondence from R to R.
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