Algebra, part 2 (tablets) - download pdf or read online

By S. B. Kizlik

ISBN-10: 1572229225

ISBN-13: 9781572229228

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Sample text

This extended set denoted by R is called the set of real numbers. We have already used the term Euclidean line without any explanation. We use the letter R (or R1 ) to denote an ordinary geometric straight line whose points have been identified with the set R of real numbers. We use the same letter R to denote the real line and the set of real numbers and say that a real number corresponds to a unique point on the line and conversely.

There is a question arising naturally: does there exist any transfinite cardinal number greater than c? The answer is affirmative. The following theorem prescribes a definite method for obtaining a set, whose cardinal number is greater than the cardinal number of the given set. 7 If A is a non-empty set and P(A) is its power set, then |A| < |P(A)|. Proof The map f : A → P(A) defined by f (a) = {a}, for all a ∈ A is injective. Hence, |A| ≤ |P(A)|. But there is no bijection g : A → P(A) (see Ex. 8 of SE-I).

Xn , . }, say. xn1 xn2 xn3 . . , where xni ∈ {0, 1, 2, . . 119999 . . ). x11 x12 x13 . . x21 x22 x23 . . .. .. . . xn1 xn2 xn3 . . .. .. . . y1 y2 y3 . . yn . . , where yn is taken to be a non-zero digit different from xnn for n = 1, 2, 3, . . , does not enter in X, since y differs from every xn at the nth place after decimal. But y determines a real number ∈ X. This contradicts that every real number r ∈ X occurs in the sequence {x1 , x2 , . . , xn , . }. Consequently, X must be non-countable.

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Algebra, part 2 (tablets) by S. B. Kizlik


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