Basic Logic Gates and De Morgan's Theorem

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