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Basic Logic Gates and De Morgan's Theorem

Basic Logic Gates and De Morgan's Theorem. Discussion D5.1 Appendix D. Basic Logic Gates and Basic Digital Design. NOT, AND, and OR Gates NAND and NOR Gates XOR and XNOR Gates DeMorgan’s Theorem. NOT Gate -- Inverter. Y. X. 0 1. 1 0. Behavior:

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Basic Logic Gates and De Morgan's Theorem

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  1. Basic Logic Gatesand De Morgan's Theorem Discussion D5.1 Appendix D

  2. Basic Logic Gates and Basic Digital Design • NOT, AND, and OR Gates • NAND and NOR Gates • XOR and XNOR Gates • DeMorgan’s Theorem

  3. NOT Gate -- Inverter Y X 0 1 1 0 Behavior: The output of a NOT gate is the inverse (one’s complement) of the input

  4. NOT • Y = ~X (Verilog) • Y = !X (ABEL) • Y = not X (VHDL) • Y = X’ • Y = X • Y = X (textook) • not(Y,X) (Verilog)

  5. NOT X ~X ~~X = X X ~X ~~X 0 1 0 1 0 1

  6. AND Gate AND X Y Z 0 0 0 0 1 0 1 0 0 1 1 1 X Z Y Z = X & Y

  7. AND • X & Y (Verilog and ABEL) • X and Y (VHDL) • X Y • X Y • X * Y • XY (textbook) • and(Z,X,Y) (Verilog) V U

  8. OR Gate OR X Y Z 0 0 0 0 1 1 1 0 1 1 1 1 X Z Y Z = X | Y

  9. OR • X | Y (Verilog) • X # Y (ABEL) • X or Y (VHDL) • X + Y (textbook) • X V Y • X U Y • or(Z,X,Y) (Verilog)

  10. Y X 0 1 1 0 Summary of Basic Gates Y = ~X not(Y,X) X Y NOT X Y Z 0 0 0 0 1 0 1 0 0 1 1 1 Z = X & Y and(Z,X,Y) X AND Z Y X Y Z 0 0 0 0 1 1 1 0 1 1 1 1 Z = X | Y or(Z,X,Y) X OR Z Y Any logic circuit can be created using only these three gates

  11. Basic Logic Gates and Basic Digital Design • NOT, AND, and OR Gates • NAND and NOR Gates • XOR and XNOR Gates • DeMorgan’s Theorem

  12. NAND Gate NAND X Y Z 0 0 1 0 1 1 1 0 1 1 1 0 X Z Y Z = ~(X & Y) nand(Z,X,Y)

  13. NAND Gate NOT-AND X Y W Z 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0 X W Z Y W = X & Y Z = ~W = ~(X & Y)

  14. 2-Input NAND Gate NAND X Y Z 0 0 1 0 1 1 1 0 1 1 1 0 X Z Y Z = ~(X & Y) nand(Z,X,Y)

  15. NOR Gate NOR X Y Z 0 0 1 0 1 0 1 0 0 1 1 0 X Z Y Z = ~(X | Y) nor(Z,X,Y)

  16. NOR Gate NOT-OR X Y W Z 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 0 X W Z Y W = X | Y Z = ~W = ~(X | Y)

  17. 2 Input NOR Gate NOR X Y Z 0 0 1 0 1 0 1 0 0 1 1 0 X Z Y Z = ~(X | Y) nor(Z,X,Y)

  18. Basic Logic Gates and Basic Digital Design • NOT, AND, and OR Gates • NAND and NOR Gates • XOR and XNOR Gates • DeMorgan’s Theorem

  19. Exclusive-OR Gate XOR X Y Z X Z 0 0 0 Y 0 1 1 Z = X ^ Y xor(Z,X,Y) 1 0 1 1 1 0

  20. XOR • X ^ Y (Verilog) • X $ Y (ABEL) • X @ Y • xor(Z,X,Y) (Verilog)

  21. X Z Y 2-Input XOR Gate XOR X Y Z 0 0 0 0 1 1 1 0 1 1 1 0 Z = X ^ Y xor(Z,X,Y) Note: if Y = 0, Z = X if Y = 1, Z = ~X Therefore, an XOR gate can be used as a controlled inverter

  22. Exclusive-NOR Gate XNOR X Y Z X Z 0 0 1 Y 0 1 0 Z = ~(X ^ Y) Z = X ~^ Y xnor(Z,X,Y) 1 0 0 1 1 1

  23. XNOR • X ~^ Y (Verilog) • !(X $ Y) (ABEL) • X @ Y • xnor(Z,X,Y) (Verilog)

  24. 2-Input XNOR Gate XNOR X Y Z 0 0 1 0 1 0 1 0 0 1 1 1 X Z Y Z = ~(X ^ Y) Z = X ~^ Y xnor(Z,X,Y) Note: Z = 1 if X = Y Therefore, an XNOR gate can be used as an equality detector

  25. Basic Logic Gates and Basic Digital Design • NOT, AND, and OR Gates • NAND and NOR Gates • XOR and XNOR Gates • DeMorgan’s Theorem

  26. NAND Gate X Z X Z = Y Y Z = ~(X & Y) Z = ~X | ~Y X Y W Z 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0 X Y ~X ~Y Z 0 0 1 1 1 0 1 1 0 1 1 0 0 1 1 1 1 0 0 0

  27. De Morgan’s Theorem-1 ~(X & Y) = ~X | ~Y • NOT all variables • Change & to | and | to & • NOT the result

  28. NOR Gate X X Z Z Y Y Z = ~(X | Y) Z = ~X & ~Y X Y Z 0 0 1 0 1 0 1 0 0 1 1 0 X Y ~X ~Y Z 0 0 1 1 1 0 1 1 0 0 1 0 0 1 0 1 1 0 0 0

  29. De Morgan’s Theorem-2 ~(X | Y) = ~X & ~Y • NOT all variables • Change & to | and | to & • NOT the result

  30. De Morgan’s Theorem • NOT all variables • Change & to | and | to & • NOT the result • -------------------------------------------- • ~X | ~Y = ~(~~X & ~~Y) = ~(X & Y) • ~(X & Y) = ~~(~X | ~Y) = ~X | ~Y • ~X & ~Y = ~(~~X | ~~Y) = ~(X | Y) • ~(X | Y) = ~~(~X & ~Y) = ~X & ~Y

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