طراحی مدارهای منطقی

93-92نیمسال دوم

دانشگاه آزاد اسالمی واحد پرند

دانشگاه آزاد اسالمی واحد پرندطراحی مدارهای منطقی

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Boolean Algebra

Boolean Algebra Basic mathematics needed for the study of the logic design of digital systems

George Boole developed Boolean algebra in 1847 Solve problems in mathematics

Claude Shannon first applied Boolean algebra to the design of switching circuits in 1939

Boolean Algebra

Boolean Variable Such as X or Y

Boolean Value or Constants 0 , 1

Basic Operations AND, OR, and complement (or inverse)

Boolean Algebra

Basic Operations AND, OR, and complement (inverse)

Complementation (Inversion)

Boolean Algebra

Basic Operations AND, OR, and complement (inverse)

AND

Boolean Algebra

Basic Operations AND, OR, and complement (inverse)

OR

Boolean Expressions and Truth Table

Boolean expressions

Formed by application of the basic operations to one or more variables or constants

Boolean Expressions and Truth Table

Boolean expressions

Evaluation

Boolean Expressions and Truth Table

Truth table (also called a table of combinations)

Specifies the values of a Boolean expression for every possible combination of values of the variables in the expression

2n rows for n input variables

Basic Theorems

Involve single variable

Commutative, Associative and Distributive laws

Commutative (جا به جایی)

Associative (شرکت پذیری)

Distributive (توزیعی)

XY = YX X+Y = Y+X

(XY)Z = X(YZ) = XYZ(X+Y)+Z = X+(Y+Z) = X+Y+Z

X(Y+Z) = XY + XZX + YZ = (X+Y)(X+Z)

Logic Optimization

A

B

C

F

AB

C

G

F=A’ + B•C’ + A’•B’

G=A’ + B•C’

Simplification Theorems

Multiplying out and Factoring

Multiplying out • Forming SOP Sum Of Products

Factoring • Forming POS Products Of Sum

DeMorgan’s Law

• DeMorgan’s Laws

• Proof

• Generalized Laws

DeMorgan’s Law

• DeMorgan’s Laws

• Example

Dual

• Replacing AND with OR, OR with AND• Replacing 0 with 1, 1 with 0 • Variables and complements are left unchanged

Exclusive-OR XOR

Exclusive-OR XOR

• Theorems

• Proof of distribution law

Equivalence Exclusive-NOR XNOR

Equivalence Exclusive-NOR XNOR

• Example

Consensus Theorem ( قانون (اجماع

• Theorem

• Proof

• Dual

Algebraic Simplification

Combining terms• XY + XY’ = X

Eliminating terms• X + XY = X

Eliminating literals• X + X’Y = X+Y

Algebraic Simplification

Example

Proving Validity of an Equation

1. Construct a truth table and evaluate both sides

2. Manipulate one side of the equation by applying various theorems until it is identical with the other side

3. Reduce both sides of the equation independently to the same expression

4. It is permissible to perform the same operation on both sides of the equation provided that the operation is reversible. For example, it is all right to complement both sides of the equation

Proving Validity of an Equation

Example