ECE 275: Digital Design I

  1. F = XY’ + XZ + X’Z’ + XYZ’ + X’Y’Z F = X((YZ') + Y') + XZ + X’Z’ + X’Y’Z -- Commutative F = X(Y' + Z') + XZ + X’Z’ + X’Y’Z -- Elimination F = X(Z + Z' + Y') + X’Z’ + X’Y’Z -- Commutative F = X(1 + Y') + X’Z’ + X’Y’Z -- Complementarity F = X + X’Z’ + X’Y’Z -- First Law F = X + Z' + X'Y'Z -- Elimination F = X + X'(Y'Z) + Z' -- Rewrite F = X + Y'Z + Z' -- Elimination F = X + Y' + Z' -- Elimination

  2. F = XYZ + X'YZ + XY'Z + XYZ' F = YZ(X + X') + XZ(Y + Y') + XY(Z + Z') -- Distribution F = YZ + XZ + XY -- Complementarity F = X(Z + Y) + YZ -- Distribution

  3. W’XY + WZ = (W’ + Z)(W + XY)

  4. W X Y Z W’XY + WZ (W’ + Z)(W + XY)
    0 0 0 0 0 0
    0 0 0 1 0 0
    0 0 1 0 0 0
    0 0 1 1 0 0
    0 1 0 0 0 0
    0 1 0 1 0 0
    0 1 1 0 1 1
    0 1 1 1 1 1
    1 0 0 0 0 0
    1 0 0 1 1 1
    1 0 1 0 0 0
    1 0 1 1 1 1
    1 1 0 0 0 0
    1 1 0 1 1 1
    1 1 1 0 0 0
    1 1 1 1 1 1
    1. F = AB’C + (A’ + B + D)(ABD’ + B’) F = AB’C + (A’ + B + D)(AD’ + B’) -- Elimination F = AB’C + ((A’ + B + D)' + (AD’ + B’)')' -- DeMorgans F = AB’C + ((AB'D') + (AD’ + B’)')' -- DeMorgans F = AB’C + ((AB'D') + ((A' + D)’ + B’)')' -- DeMorgans F = AB’C + ((AB'D') + ((A' + D) * B))' -- DeMorgans F = AB’C + ((AB'D') + (A'B + BD))' -- Distribution F = AB’C + ((AB'D' + A'B) + (AB'D' + BD))' -- Distribution F = AB’C + (((AB'D')'(A'B)') + ((AB'D')'(BD)'))' -- DeMorgans F = AB’C + (((A' + B + D)(A + B'))' + ((A' + B + D)(B' + D'))')' -- DeMorgans F = AB’C + (A' + B + D)(A + B')(A' + B + D)(B' + D') -- DeMorgans F = AB’C + (A' + B + D)(A + B')(B' + D') -- Idempotent F = AB’C + (A' + B + D)(AB' + B'B' + AD' + B'D') -- Distribution F = AB’C + (A' + B + D)(AB' + B' + AD' + B'D') -- Idempotent F = AB’C + (A' + B + D)(AB' + B' + AD' + B') -- Elimination F = AB’C + (A' + B + D)(AB' + B' + AD') -- Idempotent F = AB’C + (A' + B + D)(B' + B' + AD') -- Elimination F = AB’C + (A' + B + D)(B' + AD') -- Idempotent F = AB’C + (A'B' + BB' + B'D + AA'D' + ABD' + ADD') -- Distribution F = AB’C + A'B' + B'D + ABD' -- Complementarity F = B’(AC + A') + B'D + ABD' -- Distribution F = B’(C + A') + B'D + ABD' -- Elimination F = B’(C + A' + D) + ABD' -- Distribution F = (B’(C + A' + D) + ABD')' -- Make Complementary F = (B’(C + A' + D))'(ABD')' -- DeMorgans F = (B + (C + A' + D)')(A' + B' + D) -- DeMorgans F = (B + C'AD')(A' + B' + D) -- DeMorgans F = (A' + B' + D)B + (A' + B' + D)C'AD' -- Distribution F = A'B + B'B + DB + A'C'AD' + B'C'AD' + DC'AD' -- Distribution F = A'B + BD + AB'C'D' -- Complement