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طراحی مدارهای منطقی

طراحی مدارهای منطقی. دانشگاه آزاد اسلامی واحد پرند. نیمسال دوم 92-93. طراحی مدارهای منطقی. دانشگاه آزاد اسلامی واحد پرند. جبر بول. Boolean Algebra. Boolean Algebra  B asic mathematics needed for the study of the logic design of digital systems

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طراحی مدارهای منطقی

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

  2. طراحی مدارهای منطقی دانشگاه آزاد اسلامی واحد پرند جبر بول

  3. 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

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

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

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

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

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

  9. Boolean Expressions and Truth Table • Boolean expressions • Evaluation

  10. 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

  11. Basic Theorems • Involve single variable

  12. 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)

  13. C B G A Logic Optimization C F A F=A’ + B•C’ + A’•B’ B G=A’ + B•C’

  14. Simplification Theorems

  15. Multiplying out and Factoring • Multiplying out • Forming SOP Sum Of Products • Factoring • Forming POS Products Of Sum

  16. DeMorgan’s Law • DeMorgan’s Laws • Proof • Generalized Laws

  17. DeMorgan’s Law • DeMorgan’s Laws • Example

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

  19. Exclusive-OR  XOR

  20. Exclusive-OR  XOR • Theorems • Proof of distribution law

  21. Equivalence  Exclusive-NOR  XNOR

  22. Equivalence  Exclusive-NOR  XNOR • Example

  23. Consensus Theorem (قانون اجماع) • Theorem • Proof • Dual

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

  25. Algebraic Simplification • Example

  26. Proving Validity of an Equation • Construct a truth table and evaluate both sides • Manipulate one side of the equation by applying various theorems until it is identical with the other side • Reduce both sides of the equation independently to the same expression • 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

  27. Proving Validity of an Equation Example

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