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Art of Computer Programming, Volume 4A, The: Combinatorial Algorithms, Part 1 by Donald E. Knuth

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Exercises

Image    1. [15] What is the net effect of setting xx y, yy (x & m), xx y?

2. [16] (H. S. Warren, Jr.) Are any of the following relations valid for all integers x and y? (i) x yx | y; (ii) x & yx | y; (iii) |xy| ≤ x y.

3. [M20] If x = (xn – 1 . . . x1x0)2 with xn – 1 = 1, let Image. Thus we have 0M, 1M, 2M, 3M, . . . = –1, 0, 1, 0, 3, 2, 1, 0, 7, 6, . . ., if we let 0M = –1. Prove that (x y)M < |xy| ≤ x y for all x, y ≥ 0.

    4. [M16] Let , xN = –x, xS = x + 1, and xP = x – 1 denote the complement, the negative, ...

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