# Switching Circuits and Logic Design

## Boolean algebra

| law | | dual |
| --- | --- | --- |
| operations with 0 and 1 | X + 0 = X | X • 1 = X |
| | X + 1 = 1 | X • 0 = 0 |
| idempotent laws | X + X = X | X • X = X |
| involution laws | (X')' = X |
| laws of complementarity | X + X' = 1 | X • X' = 0 |
| commutative laws | XY = YX | X + Y = Y + X |
| associative laws | (XY)Z = X(YZ) = XYZ | (X + Y) + Z = X + (Y + Z) = X + Y + Z |
| multiple var and/or | XYZ = 1 iff X = Y = Z = 1 | X + Y + Z = 0 iff X = Y = Z = 0 |
| distributive law | AND distributes over OR | OR distributes over AND |
| | X(Y + Z) = XY + XZ | ==X + YZ = (X + Y)(X + Z)== |
| DeMorgan's law | (X + Y)' = X'Y' | (XY)' = X' + Y' |
| | (X + Y + Z)' = X'Y'Z' | (XYZ)' = X' + Y' + Z' |
| Uniting | X**Y** + X**Y'** = X | (X + **Y**)(X + **Y'**) = X |
| Absorption | X + XY = X | X(X + Y) = X |
| Elimination | X + X'Y = X + Y | X(X' + Y) = XY |
| Consensus | **X**Y + <b>X'</b>Z + **YZ** = XY + X'Z | (**X** + Y)(**X'** + Z)(**Y + Z**) = (X + Y)(X' + Z) |
| multiplying out and factoring | (X + Y)(X' + Z) = XZ + X'Y | XY + X'Z = (X + Z)(X' + Y)