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

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

Excellent, and never faraway from close to fantastic. First English language version. Moscow: Mir Publishers, 1978. The terribly scarce, and aptly titled, I.M. Yaglom treatise, An strange Algebra. Translated from the Russian via I.G. Volosova. No ISBN quantity, regardless of its 1978 book in either languages. Small 8vo, revealed eco-friendly perfect-bound wraps, 128 pp. first-class, and particularly on the subject of close to nice, a average crease at backside correct of entrance hide (scan) being the one really impressive flaw. No writing or marking, inside of or out. vibrant, stout reproduction of a real rarity within the style. See test. LT20

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Now let us discuss the meaning of the "bar" operation for the other examples of Boolean algebras considered earlier. For the algebra of two elements (Example 1 on page 25) we put 0= 1 and 1= 0 It is quite evident that for any element a of this algebra (that is for a = 0 and for a = 1) we have a = a. Further, the comparison of the "addition table" and the "multiplication table" compiled for the numbers 0 = 1 and 1 = 0 which have the form + 0 1 (I 1 0 1 1 1 and ô=i 1=0 0=1 1=0 1 0 0 0 shows that a + b = ab in all the cases.

Namely, by the sum of two propositions a and b we shall mean a proposition whose truth set coincides with the sum of the truth set A of the proposition a and the truth set В of the proposition b. Let us agree to denote this new proposition by the symbol a - f b1). Since the sum of two sets is nothing but the union of all the elements contained in both sets, the sum a + b of two propositions a and b is simply the proposition "a or b" where the word "or" means that at least one of the propositions a and b (or both propositions) is true.

The proposition a can be formed from a by prefixing "it is false that". Now let us enumerate the rules of the algebra of propositions r e h t e d to the operation of forming negation: a— a a-\-a — i o= i and and aa = o i — o and аЬ — а-\-Ъ Indeed, the negation of a necessarily false proposition (for instance, " i t is false that 2 X 2 is equal to 5" or "it is false that the pupil has two heads") is always a necessarily true proposition while the negation of a necessarily true proposition (for instance, "it is false that the pupil is not yet 120 years old") is always necessarily false.

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