1. Dr. Ameria Eldosoky Discrete mathematics

2. Basic Logic Gates

3. Basic Logic Gates and Basic Digital Design
NOT, AND, and OR Gates
NAND and NOR Gates
DeMorgan’s Theorem
Exclusive-OR (XOR) Gate
Multiple-input Gates

4. Introduction to Basic Logic Gates
Hardware consists of a few simple building blocks
These are called logic gates
AND, OR, NOT, …
NAND, NOR, XOR, …
Logic gates are built using transistors
NOT gate can be implemented by a single transistor
AND gate requires 3 transistors
Transistors are the fundamental devices
Pentium consists of 3 million transistors
Compaq Alpha consists of 9 million transistors
Now we can build chips with more than 100 million transistors

5. Basic Gates Simple gates
AND OR
NOT
Functionality can be expressed by a truth table
A truth table lists output for each possible input combination

6. AND Gate
AND
X Y Z X Z Y
Z = X + Y

7. OR Gate OR
X Y Z X Z Y
Z = X | Y

8. NOT X ~X
~~X = X
X ~X ~~X

9. NOT Gate -- Inverter X Y 1 1

10. Additional useful gates
NAND
NOR
XOR
NAND = AND + NOT
NOR = OR + NOT
XOR implements exclusive-OR function
NAND and NOR gates require only 2 transistors
AND and OR need 3 transistors!

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

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

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

14. 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 X W Z Y
W = X | Y
Z = ~W = ~(X | Y)

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

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

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

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

19. 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

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

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