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Chapter 2

Chapter 2. Logic Functions and Gates. Basic Logic Functions. The three basic logic functions are: AND OR NOT. Logic Function Representation. Logic functions can be represented: algebraically using truth tables using electronic circuits. Algebraic Representation. Uses Boolean algebra.

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Chapter 2

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  1. Chapter 2 Logic Functions and Gates

  2. Basic Logic Functions • The three basic logic functions are: • AND • OR • NOT

  3. Logic Function Representation • Logic functions can be represented: • algebraically • using truth tables • using electronic circuits.

  4. Algebraic Representation • Uses Boolean algebra. • Boolean variables have two states (binary). • Boolean operators include AND, OR, and NOT.

  5. Truth Table Representation • Defines the output of a function for every possible combination of inputs. • A system with n inputs has 2n possible combinations.

  6. Electronic Circuit Representation • Uses logic gates to perform Boolean algebraic functions. • Gates can be represented by schematic symbols. • Symbols can be either distinctive-shape or rectangular-outline.

  7. Distinctive Shape Schematic Symbols • Uses different graphic representations for different logic functions. • Uses a bubble (a small circle) to indicate a logical inversion.

  8. Rectangular-Outline Schematic Symbols • All functions are shown in rectangular form with the logic function indicated by standard notation inside the rectangle. • The notation specifying the logic function is called the qualifying symbol. • Inversion is indicated by a 1/2 arrowhead.

  9. NOT Function • One input and one output. • The output is the opposite logic level of the input. • The output is the complement of the input.

  10. NOT Function Boolean Representation • Inversion is indicated by a bar over the signal to be inverted.

  11. NOT Function Electronic Circuit • Called a NOT gate or, more usually, an INVERTER. • Distinctive-shape symbol is a triangle with inversion bubble. • Rectangular-shape symbol uses “1” and the inversion 1/2 arrowhead.

  12. NOT Function Electronic Circuit

  13. AND Function • Two or more inputs, one output. • Output is HIGH only when all of the inputs are HIGH. • Output is LOW whenever any input is LOW.

  14. AND Function

  15. AND Boolean Representation • AND symbol is “•” or nothing at all.

  16. AND Function Electronic Circuit • Called an AND gate. • Distinctive-shape symbol uses AND designation. • Rectangular-shape symbol use “&” as designator.

  17. AND Function Electronic Circuit

  18. AND Function Electronic Circuit

  19. AND Function Electronic Circuit

  20. OR Function • Two or more inputs, one output. • Output is HIGH whenever one or more input is HIGH. • Output is LOW only when all of the inputs are LOW.

  21. OR Function

  22. OR Boolean Representation • OR symbol is “+”. • Y = A + B

  23. OR Function Electronic Circuit • Called an OR gate. • Distinctive-shape symbol uses OR designation. • Rectangular-shape symbol uses “” as designator.

  24. OR Function Electronic Circuit

  25. Active Level • The logic level defined as “ON” for a circuit. • When a logic HIGH is “ON”, the signal is active-HIGH. • When a logic LOW is “ON”, the signal is active-LOW.

  26. NAND Function • Generated by inverting the output of the AND function. • Output is HIGH whenever any input is LOW. • Output is LOW only when all inputs are HIGH.

  27. NAND Function

  28. NAND Boolean Representation • Uses AND with an inversion overbar.

  29. NAND Function Electronic Circuit • Called a NAND gate. • Uses the AND symbol with inversion on.

  30. NAND Function Electronic Circuit

  31. NOR Function • Generated by inverting the output of the OR function. • Output is HIGH only when all inputs are LOW. • Outputs is LOW whenever any input is HIGH.

  32. NOR Function

  33. NOR Boolean Representation • Uses OR with an inversion overbar.

  34. NOR Function Electronic Circuit • Called a NOR gate. • Uses OR symbol with inversion on the output.

  35. NOR Function Electronic Circuit

  36. 3 Input NOR and NAND FunctionTruth Tables • 3 Input NAND: • 3 Input NOR:

  37. 3 Input NOR and NAND FunctionTruth Tables

  38. Exclusive OR Gate • Two inputs, one output. • Output is HIGH when one, and only one, input is HIGH. • Output is LOW when both inputs are equal – both HIGH or both LOW.

  39. Exclusive OR Gate

  40. Exclusive OR Gate

  41. Exclusive NOR Gate • Two inputs, one output. • Output is HIGH when both inputs are equal – both HIGH or both LOW. • Output is LOW when one, and only one, input is HIGH.

  42. Exclusive NOR Gate

  43. Exclusive NOR Gate

  44. Gate Equivalence – NAND • A NAND gate can be represented by an AND gate with inverted output. • A NAND gate can be represented by an OR gate with inverted inputs.

  45. Gate Equivalence – NAND

  46. Gate Equivalence – NOR • A NOR gate can be represented by an OR gate with inverted output. • A NOR gate can be represented by an AND gate with inverted inputs.

  47. Gate Equivalence – DeMorgan Forms • Change an AND function to an OR function and an OR function to an AND function. • Invert the inputs. • Invert the outputs.

  48. DeMorgan’s Theorem - 1 • Break the line and change the sign

  49. DeMorgan’s Theorem - 2 • The following are two common errors associated with DeMorgan’s Theorem:

  50. Active Logic Levels • Any INPUT or OUTPUT that has a BUBBLE is considered as active LOW. • Any INPUT or OUTPUT that has no BUBBLE is considered as active HIGH.

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