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Chapter 3

Chapter 3. Logic Gates & Boolean Algebra. Evaluating Expressions. Order of operation: Invert single terms Parentheses AND operations Invert result of any of the AND operations OR operations Invert result of any of the OR operations. Evaluating Expressions.

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Chapter 3

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  1. Chapter 3 Logic Gates & Boolean Algebra

  2. Evaluating Expressions Order of operation: • Invert single terms • Parentheses • AND operations • Invert result of any of the AND operations • OR operations • Invert result of any of the OR operations

  3. Evaluating Expressions We can plug in values into an expression Example: Evaluate the following expression, where A = 0, B = 1, C = 1, D = 1 X = ABC(A + D) X = 011(0 + 1) substitute the values X = 111(0 + 1) apply the inverter X = 111(1) OR first – inversion is over entire term X = 111(0) invert the parenthesis X = 1110 parenthesis means AND so not needed now X = 0 1 and 1 and 1 and 0 = 0

  4. Evaluating Logic Circuits 1 0 1 1 1 1 0 1 0 1 1 0 1

  5. Creating Circuits From Expressions AC + BC + ABC Last operation to be performed in the equation: The OR of 3 terms + BC + ABC AC

  6. Creating Circuits From Expressions Next operation to be performed in the equation: The 3 AND terms A B C AC A C BC B C A ABC B C

  7. Creating Circuits From Expressions Last operation to be performed in the equation: The 2 Inversions A B C C A AC + BC + ABC

  8. You Try This One! (D + (A + B)C) E A B C D E

  9. You Try This One! (D + (A + B) C) A B C D E

  10. You Try This One! (A + B) C) A B C D E

  11. You Try This One! (A + B) C) A B C D E

  12. You Try This One! (A + B) A B C D E (D + (A + B)C) E

  13. NOR Gates same as: A + B Expression: Truth Table: A B A + B A + B 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 0

  14. NOR Gates Note: Two bars over the same expression cancel each other A + B = A + B

  15. NOR Gates Note: Bars must be over the entire expression A + B = A + B A B X 0 0 0 0 1 0 1 0 0 1 1 1 A B X 0 0 0 0 1 1 1 0 1 1 1 1

  16. NAND Gates same as: AB Expression: Truth Table: A B AB AB 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0

  17. NAND / NOR Gate Example AB(C + D) C D A B

  18. Boolean Theorems Theorem #1: X 0 = 0 A X Z 0 0 0 0 1 0 1 0 0 1 1 1

  19. Boolean Theorems Theorem #2: X 1 = X A X Z 0 0 0 0 1 0 1 0 0 1 1 1

  20. Boolean Theorems Theorem #3: X X = X A X Z 0 0 0 0 1 0 1 0 0 1 1 1

  21. Boolean Theorems Theorem #4: X X = 0 A X Z 0 0 0 0 1 0 1 0 0 1 1 1

  22. Boolean Theorems Theorem #5: X + 0 = X A X Z 0 0 0 0 1 1 1 0 1 1 1 1

  23. Boolean Theorems Theorem #6: X + 1 = 1 A X Z 0 0 0 0 1 1 1 0 1 1 1 1

  24. Boolean Theorems Theorem #7: X + X = X A X Z 0 0 0 0 1 1 1 0 1 1 1 1

  25. Boolean Theorems Theorem #8: X + X = 1 A X Z 0 0 0 0 1 1 1 0 1 1 1 1

  26. Boolean Theorems Theorem #9: X + Y = Y + X Commutative Laws The order of the variables is unimportant. Theorem #10: X Y = Y X

  27. Boolean Theorems Theorem #11: X + (Y + Z) = (X + Y) + Z = X + Y + Z Associative Laws The grouping of like terms is unimportant Theorem #12: X(YZ) = (XY)Z = XYZ

  28. Boolean Theorems Theorem #13: X(Y + Z) = XY + XZ Distributive Law Expanding expressions is done the same way as with ordinary algebra.

  29. Boolean Theorems The distributive law also allows us to factor. Example: = B(AC + AC) ABC + ABC

  30. Boolean Theorems Theorem #14: X + XY = X Proof by factoring: X + XY = X(1 + Y) X(1 + Y) = X(1) Thm 6: X + 1 = 1 X(1) = X Thm 2: X(1) = X

  31. Boolean Theorems Theorem #15: X + XY = X + Y Variation: Let X represent X X + XY so…. X + XY = X + Y

  32. Boolean Theorems X + XY = X + Y Proof: X + XY = X + Y X0 0 1 1 Y 0 1 0 1 X 1 1 0 0 XY 0 1 0 0 XY+X 0 1 1 1 X+Y 0 1 1 1 XY 0 0 0 1 XY+X 1 1 0 1 X+Y 1 1 0 1

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