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

The current booklet is predicated at the lecture given through the writer to senior students 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 routines on the finish of every part (the such a lot tough difficulties within the routines are marked by means of an asterisk). on the finish of the booklet are positioned solutions and tricks to a couple of the issues. The reader is suggested to resolve lots of the difficulties, if no longer all, simply because in basic terms after the issues were solved can the reader make certain he is aware the subject material of the ebook. The ebook includes 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 publication. The corresponding components of the textual content of the ebook are marked via one megastar at the start and by means of stars on the finish. notwithstanding, within the moment interpreting of the booklet you could research Sec. 7 because it includes a few fabric very important for functional functions of the speculation of Boolean algebras.

The bibliography given on the finish of the ebook lists a few books which are of use to the readers who are looking to learn the idea of Boolean algebras extra thoroughly.

The writer is thankful to S. G. Gindikin for worthy suggestion and to F. I. Kizner for the thoroughness and initiative in enhancing the booklet.

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**Example text**

For the basic operations on the elements of a Boo! \ Л rigorous definition of a Boolean algebra is stated in Appendix 117. ). We shall also sometimes refer to these operations as the Boolean addition and the Boolean multiplication. In his "Laws of Thought" which first appeared in 1854, that is more than a hundred years ago, G. Boole investigated in detail this unusual algebra. The title of G. Boole's work may first seem strange; however, after the reader has studied this book it will become clear what is the relationship between unusual algebras considered in the book and the laws of human thought.

This number contains those and only those prime factors which are contained in p and are simultaneously contained in at least one of the numbers m and n. i(i п. But exactly the same factors are contained in the number (m фр) <8> (n ® p) = ([m, p], [n, p]) ,IH1 therefore we always have (m ® n) © p = (m Ф p) ~~ {n ф p) For instance, (10 ® 14) ф 105 = [(10, 14), 105] = [2, 105] = 210 and (10 e 105) ® (14 0 105) = ([10, 105], [14, 105]) == = (210, 210) = 210 Finally, in this case the roles of the elements О and 1 of the algebra of sets are played by the smallest number 1 among the collection of numbers we deal with and by the greatest number N respectively. ~~

4 reduce to nothing but the logical (propositional) connectives "or" and "and" respectively, the "bar" operation has the sense of the negation and the laws of the propositional algebra describe the basic rules for the logical operations which all the people follow in the process of thinking. Of course, in everyday life few people t h i n k of these rules as m a t h e m a t i c a l laws of thought b u t even children freely use them. Indeed, nobody doubts t h a t to say "he is a good runner and a good jumper" is just the saine as to say "he is a good jumper and a good runner", t h a t is all the people know (although they m a y not be aware of it) t h a t propositions ab and ba have the same meaning, or, which is the same, are equivalent.