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LOGIC DESIGN AND CIRCUITS

LOGIC DESIGN AND CIRCUITS. Boolean Function Minimization to SOP and POS Res . Assist . Hale İnan. Content. Boolean Operations and Expressions Boolean Addition Boolean Multiplication Laws and Rules of Boolean Algebra Commutative Law Associative Law Distributive Law

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LOGIC DESIGN AND CIRCUITS

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  1. LOGIC DESIGN AND CIRCUITS BooleanFunctionMinimizationto SOP and POS Res. Assist. Hale İnan

  2. Content • BooleanOperationsandExpressions • BooleanAddition • BooleanMultiplication • LawsandRules of BooleanAlgebra • CommutativeLaw • AssociativeLaw • DistributiveLaw • Rules of BooleanAlgebra • De Morgan’sTheorem • Standard Forms of BooleanExpressions • SOP (Sum-of-Products) form • POS (Products-of-Sum) form • Universal Gates (NAND, NOR) • Experiment

  3. BooleanOperationsandExpressions • Variable, complementandliteralaretermsused in BooleanAlgebra. • Variable : 1 or 0 aresinglevariables. • Complement : A  A’ or B  B’ • Literal : A+B , A+B+C’

  4. BooleanAddition • Booleanaddition is equivalenttothe OR operation. • InBooleanAlgebra, a sumterm is a sum of theliterals. • Examples: • A+B • A+B’ • A’+B+C+D’

  5. BooleanMultiplication • BooleanMultiplication is equivalenttothe AND operation. • InBooleanAlgebra, a productterm is theproduct of literals. • Examples: • AB • AB’ • ABC • ABCD

  6. LawsandRules of BooleanAlgebra • CommutativeLaw: • The commutative law of addition for two variables is written as A+B = B+A. • The commutative law of multiplication for two variables is A.B = B.A.

  7. LawsandRules of BooleanAlgebra • AssociativeLaws: • The associative law of addition is written as follows for three variables :A + (B + C) = (A + B) + C • The associative law of multiplication is written as follows for three variables: A(BC) = (AB)C

  8. LawsandRules of BooleanAlgebra • DistributiveLaw: • The distributive law is written for three variables as follows: A(B + C) = AB + AC

  9. Rules of BooleanAlgebra • BasicRules of BooleanAlgebra • A + 0 = A • A + 1 = 1 • A * 0 = 0 • A * 1 = A • A + A = A • A + A’ = 1 • A * A = A • A * A’ = 0 • (A’)’ = A • A + AB = A • A + A’B = A + B • (A + B)(A + C) = A + BC

  10. De Morgan’sTheorems • Theorem – 1: The complement of a product of variables is equal to the sum of the complements of the variables. • The formula for expressing this theorem for two variables is (XY)’ = X’ + Y’

  11. De Morgan’sTheorems • Theorem – 2: The complement of a sum of variables is equal to the product of the complements of the variables. • The formula for expressing this theorem for two variables is (X + Y)’ = X’Y’

  12. De Morgan’sTheorems • Examples: Apply DeMorgan's theorems to the expressions (XYZ)’ and (X + Y + Z)’. • (XYZ)’ = X’ + Y’ + Z’ • (X + Y + Z)’ = X’ Y’ Z’

  13. Standard Forms of BooleanExpressions • TheSum of Products (SOP) Form: • AB + ABC • ABC + C’DE + B’CD’ • AB + BCD + AC • The Standard SOP Form: • Theexpression A’BC’ + AB’D + ABC’D’ is madeup of thevariables A, B, C and D but D or D’ is missingfromthefirsttermand C or C’ is missingfromthesecondterm. • A’BCD’+ABC’D+AB’CD is standard SOP expression.

  14. Standard Forms of BooleanExpressions • TheProduct of Sums (POS) Form: • (A’ + B)(A + B’ + C) • (A + B’ + C’)( C + D’ + E)(B + C + D) • (A + B’)(A + B’ + C)(A + C) • The Standard POS Form: • (A+B’+C)(A+B+D’)(A+B’+C’+D) Not standard form • (A’+B’+C+D)(A+B’+C+D)(A+B+C+D) Standard form

  15. Universal Gates (NAND, NOR) • NAND & NOR gate. • A universal gate is a gate which can implement any Boolean function without need to use any other gate type. • NAND Gate is a Universal Gate: • To prove that any Boolean function can be implemented using only NAND gates, we will show that the AND, OR, and NOT operations can be performed using only these gates.

  16. Universal Gates (NAND) • Implementing NOT gate:

  17. Universal Gates (NAND) • Implementing AND gate:

  18. Universal Gates (NAND) • Implementing OR gate :

  19. Universal Gates (NOR) • Implementing NOT gate:

  20. Universal Gates (NOR) • Implementing AND gate:

  21. Universal Gates (NOR) • Implementing OR gate :

  22. Experiment-1 • Implement of thegivenBooleanfunctionusinglogicgates in SOP form. • ( AB’ + A’B’ )  Designcircuit of thisexpressionusing NAND gates.

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