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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 = (Â + B)(Â+C) Thus, we finally obtain Â + 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.