Discrete Mathematics with Graph Theory (3rd Edition) 223

Discrete Mathematics with Graph Theory (3rd Edition) 223 -...

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Section 8.1 Step 1: Let M = O. Output the words "empty set." Step 2: for i = 1 to 2 n - 1 • replace M by M + 1; • write M = 101102 ••• IOn as an n-digit number in base 2; • for k = 1 to n, if 10k = 1, output ak. 221 Each value of i in Step 2 yields one subset of { aI, ... , an} and, when Step 2 is complete, all subsets have been output. 17. (a) [BB] i. S = 1; S = 1 + (-3)(2) = -5; S = -5 + 2(22) = + 8 = 3. ii. S = 2; S = -3 + 2(2) = 1; S = 1 + 1(2) = 3. (b) i. S = 1; S = 1 + 0(5) = 1; S = 1 + 3(5 2 ) = 1 + 75 = 76. ii. S = 3; S = 0 + 3(5)=15; S = 1 + 15(5) = 76. (c) i. S = -4; S = -4 + 5( -1) = -4 - 5 = -9; S = -9 + 6( _1)2 = + 6 = -3; S = + (-4)( _1)3 = -3 - 4( -1) = + 4 = 1. ii. S = S = 6 - 4( -1) = 10; S = 5 + 1O( -1) = -5; S = -4 - 5(-1) = (d) i. S = -7; S = -7 + 16(3) = + 48 = 41; S = 41 + 0(3 2 ) = 41; S = 41 + (-40)(3 3 ) = 41 - 40(27) = 41 - 1080 = -1039; S = -1039 + 4 ) S + 17(3 5 ) + 17(243) + 4131 = 3092. ii. S = 17; S = 0 + 17(3) = 51; S = -40 + 51(3) = 113; S = 0 + 113(3) = 339; S = 16 + 339(3) = 1033; S = + 1033(3) = 3092. 18. [BB] The first value of Sis S = an. With i = 1 in Step 2, the value of Sis an-1 + Sx = + anx. With i = 2 in Step 2, the value of Sis an-2 +Sx = + (an-1 +anx)x = an_2+an_1x+anx2. With i = 3 in Step 2, the value of Sis an + = an + an-2X + an_1x2 + anx 3 . With i = the value of Sis an- n + an-(n-1)X + an_(n_2)X 2 + . .. + n = ao + a1X + a2x 2 + ... + n as desired.
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