Friday, February 20, 2015ECE120 HW51. a) Because (x + y)’ = x’y’ So (((a’ + b’)’ + (b’ + c)’)’ + c)’ = ((ab + bc’)’ + c)’ /*De Morgan*/ = (b’ + a’c + c)’/*De Morgan*/ = bc’/*Absorption and De Morgan*/ b) Because (xy)’ = x’ + y’ So (c’a’)’ + b’a’ + bc + c’d = a + c + b’a’ + bc + c’d = c(b + 1) + a + b’a’ + c’d Because x + x’y = x + y, so = c + a + b’ + d c) A(A + B’C) + A(B’ + C) = AA + AB’C + AB’ + AC /*Distributive*/ = A(1 + B’C + B’ + C) /*Distributive*/ = A(1 + B’ + C) /*Absorption*/2. a) f = (x + y + z)(x + y + z’)(x + y’ + z’)(x’ + y’ + z)(x’ + y’ + z’) g = (x + y + z’)(x + y’ + z’)(x’ + y + z)(x’ + y’ + z) b) f = xy’z + x’yz + x’yz’ g = xyz + xy’z + x’yz’ + x’y’z’ c) f’ = xyz + xyz’ + xy’z’ + x’y’z + x’y’z’ g’ = xyz’ + xy’z’ + x’yz + x’y’z d) f + g = xy’z + x’yz’ e) fg = (x + y + z’)(x + y’ + z’)(x’ + y’ + z) f ) f = xy’z + x’yz + x’yz’ ==> 3 product terms g = xyz + xy’z + x’yz’ + x’y’z’ ==> 4 product terms g) f = (x + y + z)(x + y + z’)(x + y’ + z’)(x’ + y’ + z)(x’ + y’ + z’) ==> 5 sum terms g = (x + y + z’)(x + y’ + z’)(x’ + y + z)(x’ + y’ + z) ==> 4 sum terms1Friday, February 20, 20154. a) Prime implicantsprime implicants: A’BC’D’, B’C’D, BCD, D’, A’B’ ABD, AC’D, C’AB’, AB’D’ b) Essential prime implicantsessential prime implicants: A’BC’D’, B’C’D, BCD, A’B’, D’ AB’D’ c) SOP F = A’BC’D’ + B’C’D + BCD + AB’D’ + AC’D It’s not unique since AC’D can be replaced by ABD2Friday, February 20, 2015 G = A’B’ + D’ It’s unique since they are all essential prime implicants.5. a) g = (CD + B)’(CD + (A + B)’) + (C + BD + BA)’ = (CD)’B’(CD + A’B’) + C’(BD + BA)’ /* De Morgan */ = (C’ + D’)(B’CD + A’B’) + C’(BD)’(BA)’ /* Complementarity and De Morgan */ = (A’B’C’ + A’B’D’) + C’(B’ + D’)(B’ + A’) /* Complementarity and De Morgan */ = A’B’C’ + A’B’D’ + C’(B’ + A’B’ + B’D’ + A’D’) /* Distributive */ = A’B’D’ + B’C’ + A’C’D’ /* Distributive and Absorption */ b) K-mapSOP = A’B’D’ + B’C’ + A’C’D’ c) 3 1 1 0 1 1 0 0 0 0 0 0 0 1 1 0 0Friday, February 20, 2015 d) K-mapPOS = (A + B)(C + D)(B + D)(B + C)(A + C) e) 6. a) 4 1 1 0 1 1 0 0 0 0 0 0 0 1 1 0