Basic Logic Gatesand 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

X Y

01

10

X Y

Y

NOTX Y

Y = ~X

NOT

Behavior:The output of a NOT gate is the inverse (one’s complement) of the input

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

NOT

NOT

X ~X ~~X = X

X ~X ~~X0 1 01 0 1

AND GateAND

X

Y

Z

Z = X & Y

X Y Z0 0 00 1 01 0 01 1 1

• X & Y (Verilog and ABEL)• X and Y (VHDL)• X Y• X Y• X * Y• XY (textbook)• and(Z,X,Y) (Verilog)

AND

U

V

OR Gate

OR

X

YZ

Z = X | Y

X Y Z0 0 00 1 11 0 11 1 1

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)

Y = ~Xnot(Y,X)

Summary of Basic Gates

NOT

X Y

01

10

X

YZ

X Y

X Y

Z

AND

OR

X Y Z0 0 00 1 01 0 01 1 1

X Y Z0 0 00 1 11 0 11 1 1

Z = X & Yand(Z,X,Y)

Z = X | Yor(Z,X,Y)

Any logic circuit can be created using only these three gates

Basic Logic Gates and Basic Digital Design

• NOT, AND, and OR Gates

• NAND and NOR Gates

• XOR and XNOR Gates

• DeMorgan’s Theorem

NAND GateNAND

X

Y

Z

X Y Z0 0 10 1 11 0 11 1 0

Z = ~(X & Y)nand(Z,X,Y)

NAND Gate

NOT-AND

X

Y

Z

W = X & Y

Z = ~W = ~(X & Y)

X Y W Z0 0 0 10 1 0 11 0 0 11 1 1 0

W

2-Input NAND GateNAND

X

YZ

Z = ~(X & Y)nand(Z,X,Y)

X Y Z0 0 10 1 11 0 11 1 0

NOR Gate

NOR

X

YZ

X Y Z0 0 10 1 01 0 01 1 0

Z = ~(X | Y)nor(Z,X,Y)

NOR Gate

NOT-OR

X

Y

W = X | Y

Z = ~W = ~(X | Y)

X Y W Z0 0 0 10 1 1 01 0 1 01 1 1 0

ZW

2 Input NOR Gate

NOR

XY

Z

Z = ~(X | Y)nor(Z,X,Y)

X Y Z0 0 10 1 01 0 01 1 0

Basic Logic Gates and Basic Digital Design

• NOT, AND, and OR Gates

• NAND and NOR Gates

• XOR and XNOR Gates

• DeMorgan’s Theorem

Exclusive-OR Gate

X Y ZXOR

X

YZ 0 0 0

0 1 11 0 11 1 0

Z = X ^ Yxor(Z,X,Y)

XOR

• X ^ Y (Verilog)• X $ Y (ABEL)• X @ Y

• xor(Z,X,Y) (Verilog)

X Y (textbook)

2-Input XOR Gate

XOR X Y Z0 0 00 1 11 0 11 1 0

Z = X ^ Yxor(Z,X,Y)

X Y

Z

Note: if Y = 0, Z = Xif Y = 1, Z = ~X

Therefore, an XOR gate can be used as a controlled inverter

Exclusive-NOR Gate

X Y ZXNOR

X

YZ 0 0 1

0 1 01 0 01 1 1

Z = ~(X ^ Y)Z = X ~^ Yxnor(Z,X,Y)

XNOR

• X ~^ Y (Verilog)• !(X $ Y) (ABEL)• X @ Y

• xnor(Z,X,Y) (Verilog)

X Y

2-Input XNOR Gate

XNOR X Y Z0 0 10 1 01 0 01 1 1

Z = ~(X ^ Y)Z = X ~^ Yxnor(Z,X,Y)

Note: Z = 1 if X = Y

Therefore, an XNOR gate can be used as an equality detector

XY

Z

Basic Logic Gates and Basic Digital Design

• NOT, AND, and OR Gates

• NAND and NOR Gates

• XOR and XNOR Gates

• DeMorgan’s Theorem

NAND Gate

X

Y

X

Y

Z Z

Z = ~(X & Y) Z = ~X | ~Y

=

X Y W Z0 0 0 10 1 0 11 0 0 11 1 1 0

X Y ~X ~Y Z0 0 1 1 10 1 1 0 11 0 0 1 11 1 0 0 0

De Morgan’s Theorem-1

~(X & Y) = ~X | ~Y

• NOT all variables• Change & to | and | to &• NOT the result

NOR Gate

X

YZ

Z = ~(X | Y)

X Y Z0 0 10 1 01 0 01 1 0

X

YZ

Z = ~X & ~Y

X Y ~X ~Y Z0 0 1 1 10 1 1 0 01 0 0 1 01 1 0 0 0

De Morgan’s Theorem-2

~(X | Y) = ~X & ~Y

• NOT all variables• Change & to | and | to &• NOT the result

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