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Published byRoberta Morton Modified over 4 years ago

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

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طراحی مدارهای منطقی دانشگاه آزاد اسلامی واحد پرند جبر بول

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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

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Boolean Algebra Boolean Variable Such as X or Y Boolean Value or Constants 0, 1 Basic Operations AND, OR, and complement (or inverse)

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Boolean Algebra Basic Operations AND, OR, and complement (inverse) Complementation (Inversion)

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Boolean Algebra Basic Operations AND, OR, and complement (inverse) AND

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Boolean Algebra Basic Operations AND, OR, and complement (inverse) OR

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Boolean Expressions and Truth Table Boolean expressions Formed by application of the basic operations to one or more variables or constants

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Boolean Expressions and Truth Table Boolean expressions Evaluation

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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

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Basic Theorems Involve single variable

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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)

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Logic Optimization A B C F A B C G F=A’ + B C ’ + A ’ B ’ G=A’ + B C ’

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Simplification Theorems

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Multiplying out and Factoring Multiplying out Forming SOP Sum Of Products Factoring Forming POS Products Of Sum

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DeMorgan’s Law DeMorgan’s Laws Proof Generalized Laws

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DeMorgan’s Law DeMorgan’s Laws Example

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

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Exclusive-OR XOR

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Theorems Proof of distribution law

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Equivalence Exclusive- NOR XNOR

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Example

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Consensus Theorem (قانون اجماع) Theorem Proof Dual

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Algebraic Simplification Combining terms XY + XY’ = X Eliminating terms X + XY = X Eliminating literals X + X’Y = X+Y

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Algebraic Simplification Example

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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

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Proving Validity of an Equation Example

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