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Calculus

By Daniel A. Murray

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The union of two or more open intervals is also open. A closed interval [a, b] is not an open set because the endpoints a and b are not interior points of the interval. Examples of open sets in the plane are: the interior of a disk; the Cartesian product of two one-dimensional open intervals. The reader should be cautioned that an open interval in R1 is no longer an open set when it is considered as a subset of the plane. In fact, no subset of R1 (except the empty set) can be open in R2, because such a set cannot contain a 2-ball.

In fact, we have both A s B and B c A if, and only if, A and B have the same elements. In this case we shall call the sets A and B equal and we write A = B. If A and B are not equal, we write A : B. If A c B but A B, then we say that A is a proper subset of B. It is convenient to consider the possibility of a set which contains no elements whatever; this set is called the empty set and we agree to call it a subset of every set. The reader may find it helpful to picture a set as a box containing certain objects, its elements.

Then, since i # 0, we must have either i > 0 or i < 0, by Axiom 6. Let us assume i > 0. Then taking, x = y = i in Axiom 8, we get i2 > 0, or -1 > 0. Adding 1 to both sides (Axiom 7), we get 0 > 1. On the other hand, applying Axiom 8 to -1 > 0 we find 1 > 0. Thus we have both 0 > 1 and 1 > 0, which, by Axiom 6, is impossible. Hence the assumption i > 0 leads us to a contradiction. ] A similar argument shows that we cannot have i < 0. Hence the complex numbers cannot be ordered in such a way that Axioms 6, 7, and 8 will be satisfied.

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A first course in infinitesimal calculus by Daniel A. Murray


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