ECE 275: Digital Design I
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F = XYâ + XZ + XâZâ + XYZâ + XâYâZ F = X((YZ') + Y') + XZ + XâZâ + XâYâZ -- Commutative F = X(Y' + Z') + XZ + XâZâ + XâYâZ -- Elimination F = X(Z + Z' + Y') + XâZâ + XâYâZ -- Commutative F = X(1 + Y') + XâZâ + XâYâZ -- Complementarity F = X + XâZâ + XâYâZ -- 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
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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
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WâXY + WZ = (Wâ + Z)(W + XY)
- F = ABâC + (Aâ + B + D)(ABDâ + Bâ) F = ABâC + (Aâ + B + D)(ADâ + Bâ) -- Elimination F = ABâC + ((Aâ + B + D)' + (ADâ + Bâ)')' -- DeMorgans F = ABâC + ((AB'D') + (ADâ + Bâ)')' -- DeMorgans F = ABâC + ((AB'D') + ((A' + D)â + Bâ)')' -- DeMorgans F = ABâC + ((AB'D') + ((A' + D) * B))' -- DeMorgans F = ABâC + ((AB'D') + (A'B + BD))' -- Distribution F = ABâC + ((AB'D' + A'B) + (AB'D' + BD))' -- Distribution F = ABâC + (((AB'D')'(A'B)') + ((AB'D')'(BD)'))' -- DeMorgans F = ABâC + (((A' + B + D)(A + B'))' + ((A' + B + D)(B' + D'))')' -- DeMorgans F = ABâC + (A' + B + D)(A + B')(A' + B + D)(B' + D') -- DeMorgans F = ABâC + (A' + B + D)(A + B')(B' + D') -- Idempotent F = ABâC + (A' + B + D)(AB' + B'B' + AD' + B'D') -- Distribution F = ABâC + (A' + B + D)(AB' + B' + AD' + B'D') -- Idempotent F = ABâC + (A' + B + D)(AB' + B' + AD' + B') -- Elimination F = ABâC + (A' + B + D)(AB' + B' + AD') -- Idempotent F = ABâC + (A' + B + D)(B' + B' + AD') -- Elimination F = ABâC + (A' + B + D)(B' + AD') -- Idempotent F = ABâC + (A'B' + BB' + B'D + AA'D' + ABD' + ADD') -- Distribution F = ABâC + A'B' + B'D + ABD' -- Complementarity F = Bâ(AC + A') + B'D + ABD' -- Distribution F = Bâ(C + A') + B'D + ABD' -- Elimination F = Bâ(C + A' + D) + ABD' -- Distribution F = (Bâ(C + A' + D) + ABD')' -- Make Complementary F = (Bâ(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
W | X | Y | Z | WâXY + WZ | (Wâ + 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 |