An Unusual Algebra by I.M. Yaglom, I.G. Volosova

By I.M. Yaglom, I.G. Volosova

The current ebook is predicated at the lecture given by way of the writer to senior scholars in Moscow at the twentieth of April of 1966. the excellence among the cloth of the lecture and that of the booklet is that the latter contains workouts on the finish of every part (the such a lot tough difficulties within the workouts are marked by way of an asterisk). on the finish of the e-book are put solutions and tricks to a couple of the issues. The reader is suggested to resolve lots of the difficulties, if now not all, simply because merely after the issues were solved can the reader make certain he is familiar with the subject material of the booklet. The booklet comprises a few non-compulsory fabric (in specific, Sec. 7 and Appendix that are starred within the desk of contents) that may be passed over within the first examining of the e-book. The corresponding elements of the textual content of the ebook are marked by way of one big name at the start and by way of stars on the finish. notwithstanding, within the moment interpreting of the ebook you need to research Sec. 7 because it comprises a few fabric vital for useful purposes of the idea of Boolean algebras.
The bibliography given on the finish of the e-book lists a few books which might be of use to the readers who are looking to research the speculation of Boolean algebras extra thoroughly.
The writer is thankful to S. G. Gindikin for worthwhile recommendation and to F. I. Kizner for the thoroughness and initiative in modifying the booklet.

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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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