Problem 1 (a) f(a,b,c,d) = a' b c d' + a' b c d + a b c d' + a b c' d + a b' c d + a'b'c'd' + a' b' c' d + a b' c' d' + a' b' c d + a b c d + a b c' d' + a b' c d' f(a,b,c,d) = a b + a d' + a' b' c' + b c + c d f(a,b,c,d) = (a' + b + c + d')(a + b' + c)(a + b + c' + d) (b) g(a,b,c,d) = a b'c d + a b' c' d + a b' c' d' + a b c' d' + a b c d' + a' b c d + a' b' c d + a' b' c' d + a'b'c'd' + a b c d g(a,b,c,d) = a b d' + b' c' + c d g(a,b,c,d) = (a + b' + d)(b' + c + d')(b + c' + d) (c) z(a,b,c,d,e,f) = a b c d e f + a b c d e f' + a b c d e'f' + a b c d' e' f' + a b c' d' e' f' + a b' c' d' e' f' + a b' c d' e f' + a b c' d e f' + a' b' c d e' f' + a b' c' d e f' + a b' c' d' e f + a b' c' d' e f + a b' c'd' e'f + a' b c d e f' + a' b' c d e'f' + a b c' d e f' + a b c d' e' f z(a,b,c,d,e,f) = a b c d e + a b c d' e' + a b c e' f' + a b' c d' e f' + a b' c' d' f + a c' d e f' + a c' d' e' f' + a' b' c d e' f' + b c d e f' z(a,b,c,d,e,f) = (a' + b + c' + e) (a + b' + e) (a + c) (a + d) (a + f') (b' + c + f') (b' + d + e') (b + c' + d' + e') (b + c' + f') (c + d' + e) (c + d' + f') (c + d + e' + f) (d' + e + f') Problem 2 a = x3 + x2 x1' + x2 x0' + x1' x0' a = (x0' + x1' + x3)(x0' + x2 + x3)(x1' + x2 + x3) b = x3 x2 + x3 x1 + x2'x0' + x1 x0' b = (x0' + x1 + x2)(x0' + x3)(x1 + x2' + x3) c = x3 + x2 x0 + x2'x0' + x1 c = (x0' + x1 + x2 + x3)(x0 + x1 + x2' + x3) d = x3 + x2 x1' + x2' x1 + x2 x0' d = (x0' + x1' + x2' + x3)(x1 + x2 + x3) e = x2 x1' x0 + x2' x0' + x2' x1 + x1 x0' + x3 e = (x0' + x1' + x2' + x3)(x0' + x1 + x2 + x3)(x0 + x1 + x2' + x3) f = x3 + x2' + x1 x0 + x1'x0' f = (x0' + x1 + x2' + x3)(x0 + x1' + x2' + x3) g = x3 + x2 + x1' + x0 g = x3 + x2 + x1' + x0 Problem 3 Fig. 2 x y z | g1 g2 g3 c s 0(0) 0(0) 0(0) | 0(1) 0(1) 0(1) 0(2) 0(2) 0(0) 0(0) 1(0) | 0(1) 0(1) 0(2) 0(3) 1(2) 0(0) 1(0) 0(0) | 0(1) 1(1) 0(1) 0(2) 1(2) 0(0) 1(0) 1(0) | 0(1) 1(1) 1(2) 1(3) 0(2) 1(0) 0(0) 0(0) | 0(1) 1(1) 0(1) 0(2) 1(2) 1(0) 0(0) 1(0) | 0(1) 1(1) 1(2) 1(3) 0(2) 1(0) 1(0) 0(0) | 1(1) 0(1) 0(1) 1(2) 0(2) 1(0) 1(0) 1(0) | 1(1) 0(1) 0(2) 1(2) 1(2) Fig. 3 a b c d e | g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 g12 g13 g14 g15 g16 g17 g18 g19 g20 0 0 0 0 0 | 0(2) 0(1) 0(2) 0(1) 0(2) 0(1) 0(2) 0(1) 0(2) 0(1) 0(3) 0(4) 0(3) 0(4) 0(3) 0(4) 0(3) 0(4) 0(3) 0(4) 0 0 0 0 1 | 0(3) 0(1) 0(2) 0(1) 1(1) 0(1) 0(2) 0(1) 0(2) 0(3) 0(4) 0(5) 0(3) 0(4) 0(5) 0(6) 0(3) 0(4) 0(3) 0(4) 0 0 0 1 0 | 0(2) 0(1) 0(2) 0(3) 0(4) 0(1) 0(2) 0(1) 1(1) 0(1) 0(3) 0(4) 0(3) 0(4) 0(5) 0(6) 0(3) 0(4) 0(5) 0(6) 0 0 0 1 1 | 1(3) 0(1) 0(2) 0(3) 1(1) 0(1) 0(2) 0(1) 1(1) 1(2) 1(4) 1(5) 0(3) 0(4) 0(5) 0(6) 0(3) 0(5) 0(6) 1(3) 0 0 1 0 0 | 0(2) 0(1) 1(1) 0(1) 0(2) 0(1) 0(2) 0(3) 0(4) 0(1) 0(3) 0(4) 0(5) 0(6) 0(3) 0(4) 0(3) 0(4) 0(5) 0(6) 0 0 1 0 1 | 0(6) 0(1) 1(1) 0(1) 1(1) 0(1) 0(2) 0(3) 0(4) 0(5) 0(7) 0(8) 0(9) 0(10) 0(11) 0(12) 0(3) 0(4) 0(5) 0(6) 0 0 1 1 0 | 0(2) 0(1) 1(1) 1(2) 1(3) 0(1) 0(2) 0(3) 1(1) 0(1) 0(2) 0(3) 0(4) 1(3) 1(4) 1(5) 0(3) 0(4) 0(5) 0(6) 0 0 1 1 1 | 1(3) 0(1) 1(1) 1(2) 1(1) 0(1) 0(2) 0(3) 1(1) 1(2) 1(4) 1(5) 1(6) 1(3) 1(4) 1(5) 0(3) 0(4) 0(5) 1(3) 0 1 0 0 0 | 0(2) 0(3) 0(4) 0(1) 0(2) 0(1) 1(1) 0(1) 0(2) 0(1) 0(3) 0(4) 0(4) 0(5) 0(3) 0(4) 0(5) 0(6) 0(3) 0(4) 0 1 0 0 1 | 0(4) 0(5) 0(6) 0(1) 1(1) 0(1) 1(1) 0(1) 0(2) 0(3) 0(5) 0(6) 0(7) 0(8) 0(9) 0(10) 0(11) 0(12) 0(3) 0(4) 0 1 0 1 0 | 0(2) 0(3) 0(4) 0(5) 0(6) 0(1) 1(1) 0(1) 1(1) 0(1) 0(3) 0(4) 0(5) 0(6) 0(7) 0(8) 0(9) 0(10) 0(11) 0(12) 0 1 0 1 1 | 1(3) 1(4) 1(5) 1(6) 1(1) 0(1) 1(1) 0(1) 1(1) 1(2) 1(4) 1(5) 1(6) 1(7) 1(8) 1(9) 1(10) 1(11) 1(12) 1(3) 0 1 1 0 0 | 0(2) 0(3) 1(1) 0(1) 0(2) 0(1) 1(1) 1(2) 1(3) 0(1) 0(3) 0(4) 0(5) 0(6) 0(3) 0(4) 0(5) 1(3) 1(4) 1(5) 0 1 1 0 1 | 1(5) 1(6) 1(1) 0(1) 1(1) 0(1) 1(1) 1(2) 1(3) 1(4) 1(6) 1(7) 1(8) 1(9) 1(10) 1(11) 1(12) 1(3) 1(4) 1(5) 0 1 1 1 0 | 0(2) 0(3) 1(1) 1(2) 1(3) 0(1) 1(1) 1(2) 1(1) 0(1) 0(3) 0(4) 0(5) 1(3) 1(4) 1(5) 1(6) 1(3) 1(4) 1(5) 0 1 1 1 1 | 1(3) 1(4) 1(1) 1(2) 1(1) 0(1) 1(1) 1(2) 1(1) 1(2) 1(4) 1(4) 1(5) 1(3) 1(4) 1(5) 1(6) 1(3) 1(4) 1(3) 1 0 0 0 0 | 1(1) 0(1) 0(2) 0(1) 0(2) 0(3) 0(4) 0(1) 0(2) 0(1) 1(4) 1(5) 0(3) 0(4) 0(3) 0(4) 0(5) 0(6) 0(2) 0(7) 1 0 0 0 1 | 1(1) 0(1) 0(2) 0(1) 1(1) 1(2) 1(3) 0(1) 0(2) 0(3) 0(4) 0(5) 0(3) 0(6) 0(7) 1(3) 1(4) 1(5) 0(2) 0(3) 1 0 0 1 0 | 1(1) 0(1) 0(2) 0(3) 0(4) 0(5) 0(6) 0(1) 1(1) 0(1) 0(11) 0(12) 0(3) 0(4) 0(5) 0(6) 0(7) 0(8) 0(9) 0(10) 1 0 0 1 1 | 1(1) 0(1) 0(2) 0(3) 1(1) 1(2) 1(3) 0(1) 1(1) 1(2) 1(4) 1(5) 0(3) 0(4) 0(5) 1(3) 1(4) 1(5) 1(6) 1(3) 1 0 1 0 0 | 1(1) 0(1) 1(1) 0(1) 0(2) 0(3) 0(4) 0(5) 0(6) 0(1) 0(9) 0(10) 0(11) 0(12) 0(3) 0(4) 0(4) 0(5) 0(7) 0(8) 1 0 1 0 1 | 1(1) 0(1) 1(1) 0(1) 1(1) 1(2) 1(3) 1(4) 1(5) 1(6) 1(8) 1(9) 1(10) 1(11) 1(12) 1(3) 1(4) 1(5) 1(6) 1(7) 1 0 1 1 0 | 1(1) 0(1) 1(1) 1(2) 1(3) 1(4) 1(5) 1(6) 1(1) 0(1) 1(10) 1(11) 1(12) 1(3) 1(4) 1(4) 1(5) 1(7) 1(8) 1(9) 1 0 1 1 1 | 1(1) 0(1) 1(1) 1(2) 1(1) 1(2) 1(3) 1(4) 1(1) 1(2) 1(4) 1(5) 1(6) 1(3) 1(4) 1(3) 1(4) 1(5) 1(6) 1(3) 1 1 0 0 0 | 1(1) 1(2) 1(3) 0(1) 0(2) 0(3) 1(1) 0(1) 0(2) 0(1) 0(5) 1(3) 1(4) 1(5) 0(3) 0(4) 0(4) 0(5) 0(3) 0(4) 1 1 0 0 1 | 1(1) 1(2) 1(3) 0(1) 1(1) 1(2) 1(1) 0(1) 0(2) 0(3) 0(5) 1(3) 1(4) 1(5) 1(6) 1(3) 1(4) 1(5) 0(3) 0(4) 1 1 0 1 0 | 1(1) 1(2) 1(3) 1(4) 1(5) 1(6) 1(1) 0(1) 1(1) 0(1) 1(12) 1(3) 1(4) 1(5) 1(6) 1(7) 1(8) 1(9) 1(10) 1(11) 1 1 0 1 1 | 1(1) 1(2) 1(3) 1(4) 1(1) 1(2) 1(1) 0(1) 1(1) 1(2) 1(4) 1(3) 1(4) 1(5) 1(6) 1(3) 1(4) 1(5) 1(6) 1(3) 1 1 1 0 0 | 1(1) 1(2) 1(1) 0(1) 0(2) 0(3) 1(1) 1(2) 1(3) 0(1) 1(6) 1(3) 1(4) 1(5) 0(3) 0(4) 0(4) 1(3) 1(4) 1(5) 1 1 1 0 1 | 1(1) 1(2) 1(1) 0(1) 1(1) 1(2) 1(1) 1(2) 1(3) 1(4) 1(6) 1(3) 1(4) 1(5) 1(6) 1(3) 1(4) 1(3) 1(4) 1(5) 1 1 1 1 0 | 1(1) 1(2) 1(1) 1(2) 1(3) 1(4) 1(1) 1(2) 1(1) 0(1) 1(6) 1(3) 1(4) 1(3) 1(4) 1(5) 1(6) 1(3) 1(4) 1(5) 1 1 1 1 1 | 1(1) 1(2) 1(1) 1(2) 1(1) 1(2) 1(1) 1(2) 1(1) 1(2) 1(4) 1(3) 1(4) 1(3) 1(4) 1(3) 1(4) 1(3) 1(4) 1(3) Fig. 4 x1 x2 x3 | w1 w2 w3 w4 w5 w6 w7 w8 w9 0(0) 0(0) 0(0) | 0(2) 0(1) 1(3) 1(2) 0(1) 1(3) 0(6) 0(5) 1(2) 0(0) 0(0) 1(0) | 1(1) 0(1) 1(5) 1(4) 0(1) 1(3) 0(8) 0(7) 1(2) 0(0) 1(0) 0(0) | 0(8) 0(1) 1(3) 1(2) 0(7) 0(2) 0(6) 0(5) 1(2) 0(0) 1(0) 1(0) | 1(1) 0(1) 0(4) 0(3) 1(8) 0(2) 1(7) 1(6) 1(2) 1(0) 0(0) 0(0) | 0(2) 0(3) 0(2) 1(2) 0(1) 1(6) 1(1) 1(4) 1(5) 1(0) 0(0) 1(0) | 1(1) 1(2) 0(2) 0(7) 0(1) 0(6) 1(1) 0(4) 0(5) 1(0) 1(0) 0(0) | 1(3) 1(4) 0(2) 1(2) 1(2) 0(2) 1(1) 0(5) 0(6) 1(0) 1(0) 1(0) | 1(1) 1(2) 0(2) 0(3) 1(2) 0(2) 1(1) 0(4) 0(5)