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 + ACA(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

• Proof by exhaustion : prove equivalence by comparing truth tables

• Ex: =

A B

0 0 1

0 1 1

1 0 1

1 1 0

A B

0 0 1

0 1 1

1 0 1

1 1 0

DeMorgan's Theorems

+ If you don't have both of A and B…you do not have A or do not have B

If you don't have either of A or B…you do not have A and you do not have B

DeMorgan's

• Break up solid bar by switching operation:

Samples

B

= B DeMorgan's

= B Commutative

= (1 + B) Distributive

= (1) 1+ anything = 1

= 1 · anything = self

Samples

·A

= ()·A DeMorgan's

= (A)·A Double negative cancel

= A·A Associative

= A·A Commutative

= A Anything self = self

Sum Of Products

• Can convert any truth table to sum of products expression– Each 1 in output is a product of inputs (AND)– Any of them work (OR them together)

X = B + A + AB

Multiple Inputs

• Each product shows EVERY input– Either raw or inverted

Ex:Each product hasA or B or C or

SOP Circuit

• Can build circuit from SOP expression:– Inputs feed layer of AND gates• Some inputs inverted first

– OR gate takes ANDs

X = B + A + AB

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 AND OR

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

• Convert + …. to – De Morgan's Theorem

SOP Circuit

• Can build circuit from SOP expression…

… but can often do better…

Simplification

• Can simplify things…

X = B + A + AB = B + A( + B) = B + A() = B + A = B + A = A + B

Multiple Outputs

• Multiple outputs– Each output separate function/circuit

IN OUT

A B X Y

0 0 0 1

0 1 0 0

1 0 0 0

1 1 1 1

Multiple Outputs

• Multiple outputs– Each output separate function/circuit

X = ABIN OUT

A B X Y

0 0 0 1

0 1 0 0

1 0 0 0

1 1 1 1

Multiple Outputs

• Multiple outputs– Each output separate function/circuit

X = AB

Y = AB +

IN OUT

A B X Y

0 0 0 1

0 1 0 0

1 0 0 0

1 1 1 1

Logisim

• Build simplified circuit from truth table…– Window Combinational Analysis

• Build truth table for circuit…– Project Analyze Circuit