BASIC CONCEPTS AND COMPLEX NUMBERS

BASIC CONCEPTS

AND COMPLEX NUMBERS

We begin by describing some intuitive ideas about sets.

A collection of objects under a rule or a property is call a set,
The underlying rule or property will determine, Whether the given object belongs to the collection or not. The objects which belong to a set are called its members or elements or occasionally its points.

The theory of sets has been systematically studied only since about 1880, when G. Cantor formulated the basic definitions and axioms.

He remarked that a set must consists of definite , "well defined" elements, that is to say If we have in mind some particular collection then we call it a "set" if, given any object we can decide whether it belongs to the collection we are interested in or whether it, doses not. This seemingly obvious property is violated by some objects, we can illustrate this by an example due to Bertrand Russell. Suppose we have a set of letters (alphabets), we call it S, then since S is itself an object, either S will not belong to (the set) S.

   We therefore, see that there are pitfalls in a deep study of set theory. We shall take it as an axiom that sets do exist and they contain well defined objects.

                The reader may feel that this is in some sense cheating. Let us look at the problem of language. If we look up a word in a dictionary, its meaning is given using other words.

             For instance, take the word "Straight", According to the dictionary, This means "Without bend", so we look up "bend" ; a "bend" is defined to be a "curve" and upon looking up "curve", We see that it is 'a line no part of which is straight" . If we put all these definitions together,We see that we obtain a circular definition ."straight" means "straight".

We adopt the same viewpoint here that a set is what we would intuitively think of as a collection of objects some.

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