By Azriel Levy

Aimed at upper-level undergraduate and graduate scholars, this article is composed of two elements: the 1st covers natural set conception, and the second one bargains with purposes and complex subject matters (point set topology, genuine areas, Boolean algebras, countless combinatorics and massive cardinals). invaluable appendix; a number of routines. 1979 variation. contains 20 figures.

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And /3 are two different real numbers, we can always numbers lying between a arid /3. e. is included in but not identical with (&). 9, REAL VARIABLES 10] 17 Algebraical operations with real numbers. 10. We now proceed to define the meaning of the elementary algebraical opera tions such as addition, as applied to real numbers in general. Addition. (i) In order to define the sum of two numbers we consider the following two classes (i) the class (c) formed by all sums c = a + b, (ii) the class (C) formed by all sums C= C in all cases.

Nothing two has is lost real by taking the equation by the least roots t if all three common 2 will remember that the roots are {-,b^/(b -ac)}/a. It is these lengths geometrically, first constructing *J(b 2 -ac). construct to easy much more elegant, though less straightforward, construction is the The reader A following. * See Ch. e. II, Misc. Exs. 22. there are two values of x for which are no such values of x. The reader will aa; 2 -f 2bx + c = 0. remember that algebra the equation is said to have two complex roots.

Tion, This depends, as in the case of addi a, A, b, and B Here we use the identity on showing that we can choose as small as we please. G - c = AB - ab so that (7 c is = (A - a) B + a (B - b). we include negative numbers within the scope of our by agreeing that, if a and /3 are positive, then Finally definition (-a)/3 (iv) = -a& Division. a(-/3) = -a, In order to define (-a) (-0) division, = a. we begin by de Con fining the reciprocal I/a of a number a (other than zero). fining ourselves in the first instance to positive numbers and sections of positive rational numbers, we define the reciprocal of a positive number a by means class (I/a).