# طراحی مدارهای منطقی نیمسال دوم 92-93 دانشگاه آزاد اسلامی واحد پرند.

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طراحی مدارهای منطقی نیمسال دوم 92-93 دانشگاه آزاد اسلامی واحد پرند

طراحی مدارهای منطقی دانشگاه آزاد اسلامی واحد پرند جبر بول

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  2 n 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 + XZ X + YZ = (X+Y)(X+Z)

Logic Optimization A B C F A B 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

Theorems Proof of distribution law

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

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