Concrete mathematics : a foundation for computer science

488 ANSWERS TO EXERCISES
2.2 This is lx]. Incidentally, the quantity ([x > 0] - [x < 01) is often called
sign(x) or Signum(x); it is +1 when x > 0, 0 when x = 0, and -1 when x < 0.
2.3 The first sum is, of course, a0 + al + a2 + a3 + a4 + as; the second is
a4+al+ao+al+a4,becausethesumisoverthevalueskEj-2,-1,0,+1,+2}.
The commutative law doesn’t hold here because the function p(k) = k2 is not
a permutation. Some values of n (e.g., n = 3) have no k such that p(k) = n;
others (e.g., n = 4) have two such k.
2.4 (a) E:=I E;=i+J Et=j+l aijk = IE:f=j xfzi+, ~“,++I aijk = ((al23 +
a1241 + a134) + a234.
( b ) x”,==, x:i,’ x/l; aijk = x”,=3 x?i ~~~~ aijk = Cl123 -k (Cl124 +
(a134 + a234)).
2.5 The same index ‘k’ is being used for two different index variables, although
k is bound in the inner sum. This is a famous mistake in mathematics
(and computer programming). The result turns out to be correct if oj = ok
foralljandk, l 0. Then
R(n) = A(n)ol+ B(n)0 + C(n)y + D(n)S. Setting R,, = 1 yields A(n) = 1.
Setting R, = (-1)” yields A(n) +2B(n) = (-l)n. Setting R, = (-1)“n
yields -B(n)+ZC(n) = (-1)“n. Setting R, = (-l)“n2 yields B(n)-ZC(n)+
2D(n) = (-l)“n2. Therefore 2D(n) = (-l)“(n2+n); the stated sum is D(n).
2.14 The suggested rewrite is legitimate since we have k = ~,~i~k 1 when
1 < k < n. Sum first on k; the multiple sum reduces to
t (2n+’ - 2j) = nln+’ - (2”+1 - 2) .
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