Circuit Simplification Truth Table  Minimized Logic Gates

Where We Are… Can represent a circuit's output with Boolean expression: AB + AC

Where We Are… We can simplify boolean expressions AB + AC A(B + C)

Where We Are… Thus we can simplify circuits AB + AC A(B + C)

Truth Table Truth table defines Boolean function – When in doubt, check the truth table…

Proof By Exhaustion AB AB

DeMorgan's Theorems

DeMorgan's Break up solid bar by switching operation:

Samples

Sum Of Products

Multiple Inputs

SOP Circuit

Implication 1 Can build circuit from SOP expression: Any circuit can be build with NOT, AND & OR

And… Can make NOT, AND & OR with only NAND NOT ANDOR

Implication 2 Any circuit can be build with NAND gates only…

Implication 2 Replace NOT with NAND

Implication 2 Put inversion bubbles out of AND & into OR – 2 negations cancel out

Implication 2

SOP Circuit Can build circuit from SOP expression… … but can often do better…

Simplification

Multiple Outputs Multiple outputs – Each output separate function/circuit INOUT ABXY

Multiple Outputs Multiple outputs – Each output separate function/circuit X = AB INOUT ABXY

Multiple Outputs INOUT ABXY

Logisim Build simplified circuit from truth table… – Window  Combinational Analysis Build truth table for circuit… – Project  Analyze Circuit