CMPUT 329 - Computer Organization and Architecture II1 CMPUT329 - Fall 2002 Topic2: DeMorgan Laws José Nelson Amaral

CMPUT 329 - Computer Organization and Architecture II

1

CMPUT329 - Fall 2002

Topic2: DeMorgan Laws

José Nelson Amaral


2

Reading Assignment

Chapter 4: Sections 4.1.4, 4.1.5, 4.1.6, 4.2, 4.3.1, 4.3.2


3

First DeMorgan’s Law

The complement of the sum is equal the product of the complements.

(X+Y)’ = X’Y’

XY

Z

X Y X+Y (X+Y)’ X’ Y’ X’Y’0 0 0 1 1 1 10 1 1 0 1 0 01 0 1 0 0 1 01 1 1 0 0 0 0

Z

Y

X


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(X+Y)’ = X’Y’

XY

Z

X Y X+Y (X+Y)’ X’ Y’ X’Y’0 0 0 1 1 1 10 1 1 0 1 0 01 0 1 0 0 1 01 1 1 0 0 0 0

Z

Y

X


5



(X+Y)’ = X’Y’

XY

Z

X Y X+Y (X+Y)’ X’ Y’ X’Y’0 0 0 1 1 1 10 1 1 0 1 0 01 0 1 0 0 1 01 1 1 0 0 0 0

Z

Y

X


6



(X+Y)’ = X’Y’

XY

Z

X Y X+Y (X+Y)’ X’ Y’ X’Y’0 0 0 1 1 1 10 1 1 0 1 0 01 0 1 0 0 1 01 1 1 0 0 0 0

Z

Y

X


7



(X+Y)’ = X’Y’

XY

Z

X Y X+Y (X+Y)’ X’ Y’ X’Y’ 0 0 0 1 1 1 1 0 1 1 0 1 0 0 1 0 1 0 0 1 0 1 1 1 0 0 0 0

Z

Y

X


8

Second DeMorgan’s Law

The complement of the product is equal the sum of the complements.

(XY)’ = X’ + Y’

ZXY

X Y XY (XY)’ X’ Y’ X’ + Y’0 0 0 1 1 1 10 1 0 1 1 0 11 0 0 1 0 1 11 1 1 0 0 0 0

Z

Y

X


9

NOR and NAND

Because these combination of gates are used often, thereare special symbols to represent them:

XY

Z XY

Z

ZXY

XY

Z


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DeMorgan’s Law (cont.)

The De Morgan’s laws generalize to n variables:

(X1 + X2 + X3 + ··· + Xn)’ = X1’X2’X3’ ··· Xn’

(X1X2X3 ··· Xn)’ = X1’ + X2’ + X3’ + ··· + Xn’


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DeMorgan’s Law (example)

Express the complement f’(w,x,y,z) of the following expression in a simplified form.

f(w,x,y,z) = wx(y’z + yz’)

f’(w,x,y,z) = w’ + x’ + (y’z +yz’)’

= w’ + x’ + (y’z)’(yz’)’

= w’ + x’ + (y + z’)(y’ + z)

= w’ + x’ + yy’ + yz + z’y’ + z’z

= w’ + x’ + 0 + yz + z’y’ + 0

= w’ + x’ + yz + y’z’


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Exclusive OR

X Y = XY’ + X’YX Y C0 0 00 1 11 0 11 1 0

If X=1 OR Y=1, butnot both of them, then C=1

XY

C

X 0 =

X 1 =

X X =

X X’ =

Commutative Law:X Y = Y X

Associative Law:(X Y) Z= X ( Y Z) = X Y Z

Distributive Law:X(Y Z) = XY XZ

X

X’

0

1


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Exclusive-OR (cont.)

Complement Law:(X Y)’ = X Y’ = X’ Y

X Y X’ Y’ XY (XY)’ XY’ X’Y0 0 1 1 0 1 1 10 1 1 0 1 0 0 01 0 0 1 1 0 0 01 1 0 0 0 1 1 1

Algebraic Proof:(X Y)’ = (XY’ + X’Y)’

= (XY’)’(X’Y)’= (X’ + Y)(X + Y’)= X’X + X’Y’ + XY + YY’= 0 + X’Y’ + XY + 0

= X’ Y= X’Y’ + XY= XY + X’Y’ = X Y’


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In-place Value Permutation

The following properties are valid:(XY)Y = X(XY)X= Y

X Y X1=XY Y1=X1Y X2=X1Y1 0 0 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1

Algebraic: (XY) Y = (XY’ + X’Y)Y’ + (XY’ + X’Y)’Y= XY’Y’ + X’YY’ + ((XY’)’(X’Y)’)Y= XY’ + 0 + ((X’+Y)•(X+Y’))Y= XY’ + X’XY + X’Y’Y +XYY + YY’Y= XY’ + 0 + 0 + XY + 0= X(Y’ + Y)= X•1= X


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Using In-place Value Permutation in Assembly

Can be used in assembly programming to exchange thevalue of two registers in place:

R1 R1R2R2 R1R2R1 R1R2

The In-place Value Permutation Property of the exclusive-OR:(XY)Y = X(XY)X= Y

If we do back substitution in the second and third operations,we will find out that (assuming R1=A and R2=B initially):R1 (A B)R2 (A B) B = AR1 (A B) A = BThus, if initially R1 = A and R2 = B, then after this sequence of operations, R1 = B and R2 = A.


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Equivalence Gate

(X Y) = XY + X’Y’X Y C0 0 10 1 01 0 01 1 1

If X=Y then C=1,otherwise C=0

(X Y) = (X Y)’

XY

C


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Positive and Negative Logic

Positive Logic: the higher voltage (+V) represents 1 and the lower voltage (0V) represents 0

Negative Logic: the higher voltage (+V) represents 0 and the lower voltage (0V) represents 1


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Positive and Negative Logic (example)

LogicGate

e2

e3

e1

eo

The same physical circuit implements different logicfunctions. The function implemented depends on the logic used to interpret the inputs and outputs.

e1 e2 e3 eo

0V 0V 0V 0V0V 0V +V 0V0V +V 0V 0V0V +V +V 0V+V 0V 0V 0V+V 0V +V 0V+V +V 0V 0V+V +V +V +V

+

Electric Voltages

e1 e2 e3 eo

0 0 0 00 0 1 00 1 0 00 1 1 01 0 0 01 0 1 01 1 0 01 1 1 1

+

Positive Logic

e1 e2 e3 eo

1 1 1 11 1 0 11 0 1 11 0 0 10 1 1 10 1 0 10 0 1 10 0 0 0

+

Negative Logic


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Duality

Any theorem or identity in switching algebraremains true if 0 and 1 are swapped and • and + are also swapped throughout.

Example:X + X • Y = X

X • X + Y = X (duality)X + Y = X (using idempotency)

what is wrong?

We must keep the operatorprecedence of the originalexpression, therefore:X + X • Y = XX • (X + Y) = X (duality)


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The Consensus Theorem

XY + X’Z + YZ = XY + X’Z

= XY + X’Z + (X + X’)YZ

= XY + X’Z + XYZ + X’YZ

= XY + XYZ + X’Z + X’YZ

= XY(1 + Z) + X’Z(1 + Y)

= XY·1 + X’Z·1

XY + X’Z + YZ = XY + X’Z + 1·YZ

= XY + X’Z


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Sum-of-Products (Minterms)

An expression is on a sum-of-products form if it isformed by the sum of products, and all the

products are formed by single variables only.

Examples:AB’ + CD’E + AC’E’ABC’ + DEFG + HA + B’ + C + D’E

The following expressions are not sum-of-products:(A+B)CD + EF(X + Y)(X + Z)

Each of the products in the sum-of-products form iscalled a minterm. Thus the form is also called

sum-of-minterms.


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Product-of-Sums (Maxterms)

Similarly, a product-of-sums is formedby the product of sums in which all the

sums are formed by single variables only.

Examples:(A + B’)(C + D’ + E)(A + C’ + E’)(A + B)(C + D +E)FAB’C(D’ + E)

The following expressions are not product-of-sums:(A+B)CD + EFA + B’ + C + D’E

Each of the sums in the product-of-sums form iscalled a maxterm. Thus the form is also called

product-of-maxterms.


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Building Practical Circuits

Problem: Design a logical circuit to automatically operatea car alarm. The manual for the alarm gives the following explanation for its operation.

“The alarm will go off if the alarm system is activated and any of the two doors or the trunk are open, or if the vibration sensor is activated and the key is not in the ignition.”


AlarmActivatedDriverDoorOpenPassengerDoorOpenTrunkOpenVibrationKeyInIgnition

Inputs:


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AlarmGoOff = AlarmActivated • (DriverDoorOpen + PassengerDoorOpen + TrunkOpen) + Vibration • (KeyInIgnition)’

AlarmActivatedDriverDoorOpenPassengerDoorOpenTrunkOpenVibrationKeyInIgnition

Inputs:


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AlarmGoOff = AlarmActivated • (DriverDoorOpen + PassengerDoorOpen + TrunkOpen) + Vibration • (KeyInIgnition)’

VibrationKeyInIgnition

AlarmGoOff

DriverDoorOpenPassengerDoorOpen

TrunkOpen

AlarmActivated


26

Building a Circuit From a Truth Table

Design a logic circuit to implement a two bit adder. This circuithas three inputs (A, B, Cin), and two outputs (S, Cout). Theoutput S is one if the sum is one, i.e., the number of inputsequal one is odd. The carry out output is one if the sum producesa carry, i.e., two or more of the inputs are one.

Adder

Cin

Cout

SB

A

A B Cin S Cout0 0 0 0 00 0 1 1 00 1 0 1 00 1 1 0 11 0 0 1 01 0 1 0 11 1 0 0 11 1 1 1 1

+


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Building a Circuit From a Truth Table

Adder

Cin

Cout

SB

A

A B Cin S Cout0 0 0 0 00 0 1 1 00 1 0 1 00 1 1 0 11 0 0 1 01 0 1 0 11 1 0 0 11 1 1 1 1

+

S = A’B’Cin + A’BCin’ + AB’Cin’ + ABCin

Cout = A’BCin + A B’Cin + ABCin’ + ABCin

= A’BCin + ABCin + AB’Cin + ABCin + ABCin’ + ABCin

= BCin + ACin + AB

= (A’ + A)BCin + (B’ + B)ACin + (Cin’ + Cin)AB

= 1·BCin + 1· ACin + 1· AB

Documents

CMPUT 329 - Computer Organization and Architecture II1 CMPUT329 - Fall 2002 Topic2: DeMorgan Laws José Nelson Amaral