. A..2
2 2. () (xyz) ( xyz) ( xy z) = (x x)yz ( xy z) = yz ( xy z) = y(z ( x z)) = y((z x)(z z)) = y( x z) (2) (3) M aj (x, y, M aj ( x, ȳ, z)) = xy ȳm aj ( x, ȳ, z) M aj ( x, ȳ, z)x M aj (x, y, z) x = xy ȳ(xȳ y z z x) ( x y)(ȳ z)( z x)x = xy xȳ xȳz ( xȳ xz yz)( z x)x = xy xȳ xȳz ( xȳ z xyz)x = xy xȳ xȳz xyz = x(y ȳz) x(ȳ yz) = x(y z) x(ȳ z) = xy xȳ xz xz = x y z = M aj (0, y, z) M aj (, y, z) = (y z z) (ȳ y z) = (y z z)(ȳ y z) (y z z)(ȳ y z) = (y z z)y(ȳ z) (ȳ z) z(ȳ y z) = (y z z)yz ȳ z(ȳ y z) = yz ȳ z
2.2 ()F (x, y, z) = x y z 3 F (x, y, z) = xyz xȳ z xy z xȳz F (x, y, z) = ( x ȳ z)(x y z)(x ȳ z)( x y z) (2)F (x, y, z) = (x y)(ȳ z) F (x, y, z) = (x y)(ȳ z) = xȳ xz yz = xyz xȳz xyz xȳ z F (x, y, z) = (xy z) ( xȳ z) ( xȳz) ( xy z) = ( x ȳ z)(x y z)(x y z)(x ȳ z) (3)F (x, y, z) = (xȳ) (y z) (x z) F (x, y, z) = (xȳ) (y z) (x z) = xȳ y z x z
4 = ( x y)(ȳ z)( x z) = xȳz xz xȳ xyz yz = xȳz xyz xȳ z xyz F (x, y, z) = ( xy z) (xȳz) (xy z) (xȳ z) = (x ȳ z)( x y z)( x ȳ z)( x y z) (4)F (x, y, z) = (x ȳ) xz F (x, y, z) = (x ȳ) xz = (x ȳ) xz (x ȳ)( xz) = xy xz ((x ȳ)(x z)) = xyz (x ȳ z) = xyz xyz xȳz xy z xȳ z xȳ z F (x, y, z) = ( xȳz) ( xy z) = (x y z)(x ȳ z)
5 2.3 () x yz = xyz xȳz xy z xȳ z () (2) (xy ȳz) z = x ȳ z () (3) x(y z) = xȳ zxyz (NAND ) (4) xȳ xz = x z y 0 x 0 y 0 z (NOR ) 2.4 () x y z x y z = F (x, y, z) F (x, y, z) = x y z = xȳ z F (0, y, z) = ȳ z F (, y, z) = 0 z = z F (0, y, z) F (, y, z) = ȳ z x z, y ȳ z = z y y, z (y, z) = (0, 0), (0, ), (, ) x α x = F (0, y, z) F (, y, z)α = 0 zα = zα (x, y, z) = (0, 0, 0), (0, 0, ), (, 0, ), (0,, ), (,, ) (2) x y = xz y (x y)(ȳ xz) = F (x, y, z)
6 F = (x y)(ȳ xz) F (0, y, z) = y(ȳ z) = yz F (, y, z) = ȳ F (0, y, z) F (, y, z) = yz ȳ = z ȳ x z, y z ȳ = z y x α x = F (0, y, z) F (, y, z)α = yz ȳα = ȳ z ȳα = ȳ z z y ȳ z x = ȳ y, z z ȳ = (y, z) = (0, 0), (0, ), (, ) (x, y, z) = (, 0, 0), (, 0, ), (0,, ) (3) x ȳz = F (x, y, z) F (x, y, z) = x ȳz F (0, y, z) = ȳz F (, y, z) = y z F (0, y, z) F (, y, z) = z, y x x α
7 x = F (0, y, z) F (, y, z)α = y z (y z)α = y z (x, y, z) = (, 0, 0), (0, 0, ), (,, 0), (,, )
8 3. F 3 (x 5, x 4, x 3, x 2, x ) F 2 (x 5, x 4, x 3, x 2, x ) AND F 3 (x 5, x 4, x 3, x 2, x ) OR F 2 (x 5, x 4, x 3, x 2, x ) 3.2 (xy ȳz) z x, y, z 0 x = 0 ȳz z = ȳ zx = x y = 0 y = x z y z = 0 z = xy ȳ z (xy ȳz) z 3.3 x, y, z 3 0,,2,3 0 0 4 3 0 0 4 xy xz yz, xȳ z xy z xȳz xyz, xyz xȳz xy z xȳ z, ȳ z x z xȳ 3.4 f(x 3, x 2, x ) = a 3 x 3 a 2 x 2 a x a 0
9 f( x 3, x 2, x ) = a 3 (x 3 ) a 2 (x 2 ) a (x ) a 0 = a 3 (x 3 ) a 2 (x 2 ) a (x ) a 0 = a 3 x 3 a 2 x 2 a x a 3 a 2 a a 0 3.5 3 f(x, y, z) ()xy z (x y) z (2)x ȳ z (3)x y z (4) f(0, 0, 0) = f(, 0, 0) = f(, 0, ) = f(,, 0) = 0 yz x(y z) (5)xyz (6)x y z 87) xy yz xz
0 x3 x2 x x5 x4 A4.. M aj (x 5, x 4, x 3, x 2, x ) x3 x2 x x5 x4 A4..2 H 2 (x 5, x 4, x 3, x 2, x ) 4. () M aj (x 5, x 4, x 3, x 2, x ) 4A.. NOT-AND-OR M aj (x 5, x 4, x 3, x 2, x ) = x 3 x 2 x x 4 x 2 x x 5 x 2 x x 4 x 3 x x 5 x 3 x x 5 x 4 x x 4 x 3 x 2 x 5 x 3 x 2 x 5 x 4 x 2 x 5 x 4 x 3 (2) H 2 (x 5, x 4, x 3, x 2, x ) H 2d (x 5, x 4, x 3, x 2, x ) ( 4 ) A4..2 A4..3 NOT-AND- OR AND OR H 2 (x 5, x 4, x 3, x 2, x ) NOT-OR-AND H 2d (x 5, x 4, x 3, x 2, x ) = x 4 x 3 x 2 x x 5 x 3 x 2 x x 5 x 4 x 2 x x 5 x 4 x 3 x x 5 x 4 x 3 x 2
x3 x2 x x5 x4 A4..3 H 2d (x 5, x 4, x 3, x 2, x ) x3 x2 x ux5 x4 A4..4 u = H 2 (x 5, x 4, x 3, x 2, x ) H 2 (x 5, x 4, x 3, x 2, x ) = (x 4 x 3 x 2 x )(x 5 x 3 x 2 x )(x 5 x 4 x 2 x ) (x 5 x 4 x 3 x )(x 5 x 4 x 3 x 2 ) (3) M aj (x 5, x 4, x 3, x 2, x ) = x 5 M aj (, x 4, x 3, x 2, x ) x 5 M aj (0, x 4, x 3, x 2, x ) (4) u = H 2 (x 5, x 4, x 3, x 2, x ) M aj (x 5, x 4, x 3, x 2, x ) A4..4 A4..4 () M aj (x 5, x 4, x 3, x 2, x )
2 M aj (x 5, x 4, x 3, x 2, x ) = x 3 x 2 x x 4 x 2 x x 5 x 2 x x 4 x 3 x x 5 x 3 x x 5 x 4 x x 4 x 3 x 2 x 5 x 3 x 2 x 5 x 4 x 2 x 5 x 4 x 3 4.2 4.0 A4..5 NOT-AND x 5 x 3 x x 5 x 3 x 2 x 5 x 4 x 3 x 5 x 2 x x 5 x 4 x 2 x 5 x 2 x x 4 x 3 x x 5 x 3 x x 4 x 2 x x 5 x 4 x 3 x 5 x 3 x x 5 x 4 x x 5 x 4 x 2 x 5 x 2 x x 4 x x 3 x 2 0 A4.2. 2 3 3 A4.2.2 2 A4.2.3 x 3 x 2, x 5 x 2 x, x 5 x 3 x, x 5 x 2 x 4?? NOT-AND-OR F (x 5, x 4, x 3, x 2, x ) = x 3 x 2 x 5 x 2 x x 5 x 3 x x 5 x 2 x 4.3 (2)y y 2 y, y 2 ( A4.3)
3 4.4 y = x 3 x 2 x 3 x 2 y 2 = x 2 x x 3 x 2 ()(2)(3) NOT AND OR NAND 4.5 A4.4 NOT-AND-OR f(x, y, z) = xȳ z xz A4.4 NAND 3 H = x, H 2 = z τ = ȳ z, tau 2 x H τ H 2 tau 2 f(x, y, z) = xȳz z x = xȳxz zxz xz NAND xz NAND f(x, y, z) = xȳ zzx = xȳxzzxz 3 x 3, x 2, x u = x x 2 x 3 u A4.5 y = u x 3 u x 2
4 p = 0 0 0 0 0 0 p = 0 0 0 0 0 0 0 0 0 0 0 0 p = 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 p = 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 p = 4 0 0 0 0 p = 5 (a) p = 0 0 0 0 0 0 0 0 0 0 0 0 0 p = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 p = 2 0 0 0 0 0 0 0 0-4 3-2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 p = 3 0 0 0 0 0 0 0 0 0 0 0 0 0 p = 4 (b) A4..5 NOT-AND p = 0 0 0 0 0 0 0 0 0 0 p = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 p = 2 0 0 0 0 0 0 0 0 0 0 0 p = 3 (c) p = 0 p = 2 (d)
5 A4.2. A4.2.2 3
6 A4.2.3 2 x2 x x2 x x2 x x3 x3 x3 A4.3 y, y 2, y y 2 y, y 2
7 A4.4 NOTANDOR NAND NAND x2 x u x3 A4.5