By Brahmagupta

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Extra resources for Algebra, with Arithmetic and mensuration,: From the Sanscrit of Brahmegupta and BhaÌscara

Example text

Indeed, for any number x satis( х ® у ; ф z=(x®zmy@z) x®z = y®z x®y oi—o-!. у х о Z ii oi—о (a) о X (b) Fig. 14 30 Z (x®y)®z=(x®z)®(y®z) x®y=x®z У®г о y II the condition 0 ^ x ^ 1 we always have x © 0 = max [x, 0] = x and x ® 1 = min [x, 1] = x x Ф 1 — max [x, 1] = 1 and x ® 0 = min [x, 0] = 0 lying Example 4. Algebra of least common multiples and greatest common divisors. Let N be an arbitrary integer. As the elements of the new algebra we shall take all the possible divisors of the number N. For instance, if N = 210 = = 2 -3 -5 -7 then the elements of the algebra in question are the numbers 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105 and 210.

Similarly, the negation of the proposition "the pupil has no bad marks" means thai "the pupil has bad^marks" and the double negation of the former proposition slates that "it is falsi' llial the pupil has bad marks" and is therefore equivalent to the original proposition asserting t h a t the pupil has no bad marks. The De Morgan rules a-\-b — ab ab-=a-\-b and for the propositions are also very i m p o r t a n t ; the verbal s t a t e m e n t of lliese rules is a little more complicated (in S h is connection see Exercise 1 below).

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. (b) Let TV = pA where p is a prime number and A is a positive integer.