Absolutely summing operators from the disc algebra

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Considered one of Che hallmarks of major advances in arithmetic is the invention of unforeseen relationships among varied parts. maybe the main amazing fresh instance of such an increase is D« Quillen! s improvement of upper algebraic K-theory within which algebra and topology are similar in a new and basic method.

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Check that in the "algebra of four numbers" (Example 2 on page 27) there holds De Morgan's second rule ab = a -)- b. 5*. (a) Let N = ргр2 . . , Ph a r e pairwise different. Prove t h a t in this case the "algebra of least common multiples and greatest common divisors" whose elements are the divisors of the number N (see Example 4 on page 31) reduces to the "algebra of the subsets of the universal set / = p2, . , ph}". Proceeding from this fact show that in this "algebra of least common multiples and greatest common divisors" all the laws of a Boolean algebra hold including the De Morgan rules.

Ow suppose that we have an arbitrary relation holding in any Boolean algebra, for instance, the equality A (A + С) (В + С) = AB + AC which we have already mentioned. The application of the "bar " operation to both members of this equality results in A{A + C) (B + C) = AB + AC However, by virtue of the De Morgan rules, we have A(A + C) (B + C) = [A(A + C)](B + C) = =-A(A + C) + B + C = I + A + C + BC = Â+ÂC + BC and AB + AC = ÂBÂC = ( + B)(Â+C) Thus, we finally obtain  + ÂC + BC = (À + B) {Â+ С) Since the last equality is fulfilled for any А , В and С it remains valid if we simply denote the elements A, В and С of the Boolean algebra by the letters А, В and С; this yields the equality A + AC + ВС = (A + В) (A + C) which is the dual of the original equality.

Is it necessarily false? 3. Let the proposition a mean "the pupil can play chess" and let the proposition b be "the pupil can play draughts". Explain the meaning of the following propositions: (a) a + b; (b) ab; (c) a + b; (e) a+Ъ; (f) ab; (g) ab; (d) a-\~T>; (h) ab 4. Let a be the proposition "he is an excellent pupil", b "he is dark" and let с mean "he can swim". Explain the meaning of the propositions (a) (a + b) с and ac -f- be and (b) ab + с and (a + c) (b + c) 5. Let the propositions a and b mean "the given positive integer is even" and "the given positive integer is a prime number" respectively.

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