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DLD Lecture 4.pptx

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DLD Lecture 4.pptx

  1. 1. Overview • Boolean Algebra o Axioms and Theorems o Principle of Duality o Boolean Operator Precedence o Boolean Function Evaluation 29-Nov-22 CSE 225: Digital Logic Design 2
  2. 2. Boolean Algebra  A branch of mathematics that deals with operations (AND, OR, NOT) on logical values with binary variables.  Based on a set of rules derived from a small number of basic assumptions - Axioms 29-Nov-22 CSE 225: Digital Logic Design
  3. 3. Axioms 29-Nov-22 CSE 225: Digital Logic Design
  4. 4. Single Variable Theorems 29-Nov-22 CSE 225: Digital Logic Design
  5. 5. Theorems 29-Nov-22 CSE 225: Digital Logic Design Commutative Associative Distributive Absorption
  6. 6. Theorems 29-Nov-22 CSE 225: Digital Logic Design Combining DeMorgan’s Theorem Consensus
  7. 7. Proof of DeMorgan’s Theorems 29-Nov-22 CSE 225: Digital Logic Design
  8. 8. Principle of Duality  The dual of an algebraic expression is obtained by interchanging + and · and interchanging 0’s and 1’s.  The identities appear in dual pairs. When there is only one identity on a line the identity is self-dual, i. e., the dual expression = the original expression.  Unless it happens to be self-dual, the dual of an expression does not equal the expression itself. 29-Nov-22 9 CSE 225: Digital Logic Design
  9. 9. • F1 = (A + C) · B + 0 Dual F1 = (A · C + B) · 1 = A · C + B • F2 = X · Y + (W + Z) Dual F2 = • F3 = A · B + A · C + B · C Dual F3 = • F4 = X · Y + Y · Z + X · Z = X · Y + X · Z Dual F4 = • F5 = X · (Y + Z) = X · Y + X · Z Dual F5 = • Are any of these functions self-dual? Principle of Duality 29-Nov-22 10 CSE 225: Digital Logic Design
  10. 10. Boolean Operator Precedence  The order of evaluation in a Boolean expression: 1. Parentheses 2. NOT 3. AND 4. OR  Consequence: Parentheses appear around OR expressions  Example: F = A(B + C)(C + D) 29-Nov-22 11 CSE 225: Digital Logic Design
  11. 11. Problem 1 29-Nov-22 12 CSE 225: Digital Logic Design
  12. 12. Problem 2 29-Nov-22 13 CSE 225: Digital Logic Design
  13. 13. Problem 3 ) Z X ( X Z ) Y X ( + = + + Y Y 29-Nov-22 14 CSE 225: Digital Logic Design
  14. 14. Boolean Function Evaluation x y z F1 F2 F3 F4 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 1 1 0 1 z x y x F4 x z y x z y x F3 x F2 xy F1 + = + = = = z yz + y + 29-Nov-22 15 CSE 225: Digital Logic Design

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