1. Boolean Algebra and Gate Networks
Modern digital computers are designed , maintained , and their operation is analyzed using techniques and symbology from a field of mathematics called modern algebra
The name Boolean algebra honors a fascinating English mathematics called modern algebra
Boolean algebra was first brought to bear on problems by Claude Shannon in 1938 (switching circuits)

2. Boolean Algebra & logic gates
In formula 2x+5y=z, in normal algebra (x,y and z may range through the entire field of real numbers)
In Boolean algebra , the variables take only one of two possible values ( 0 or 1).
So in equation x+y=z each of the variables x,y and z will be one of the values 0 or 1.

3. The possible input and output combinations may be arranged as follows:
0+0=0
0+1 =1
1+0 =1
1+1 =1
This is called logical addition (+ OR)

4. The second important operation in Boolean algebra is called (
The second important operation in Boolean algebra is called (. AND) or Logical Multiplication.
0 . 0 =0
0 .1 =0
1 .0 =0
1 . 1 =1
Both (+) and (.) Obey a mathematical rule called the associative law.
(x+y)+z = x +(y+z)
(x.y).z = x.(y.z)

5. If +’s and. ’s mixed , then the rule which is used is that (
If +’s and .’s mixed , then the rule which is used is that (. always performed before +)
So x.y+z = (x.y )+ z
And x.y + x.z means (x.y) + (x.z)

6. Logical Multiplication function Input Output A B F
The (.) Operations are physically realized by electronic circuit called AND Gate
Logical Multiplication function
Input
Output A B F 1 A
1 output
2 inputs
3 inputs
4 inputs
Multiple inputs F B

7. AND Gate Timing Diagram

8. Logical Add function Input Output A B F 1
The + Operations are physically realized by electronic circuit called OR Gate
Logical Add function
Input
Output A B F 1
1 output
2 inputs
3 inputs
4 inputs
Multiple inputs A F B

9. OR Gate Timing Diagram

10. NOT Gate A Input Output A F 1 Invert function
In Boolean algebra we have an operation called “Invertor” or “complementation” and the (- ) symbol will be used with the variable.
So X is meaning the complement of X
1 input
1 output A
Input
Output A F 1
Invert function

11. NOT Gate Timing Diagram

12. The complement of a value can be taken repeatedly
0 = 1 = 0 = 1

13. AND Gate Applications Enable/Disable Device
Counter counts when it receives pulses

14. OR Gate Applications
Car door open alarm

15. NOT Gate Applications
1’s Complement

16. Evaluation of logical expressions
To study a logical expression it is very useful to construct a table of values (Truth table) for the variables then evaluate the expression for each of the possible combination of variables in turn
Consider the expression x+yz , there are 3 variables x,y,z
so there are 2^3 combinations.
X Y Z ‘Z
Y’Z
X+Y’Z

17. expression x.y+ x.z is also 3 variables
y+z
x．(y+z)
x．y
x．z
(x．y)+(x．z) 1

18. NAND Gate function
NOT-AND function
Input
Output A B F 1 B A

19. NAND Gate Timing Diagram

20. NOR Gate function
NOT-OR function
Input
Output A B F 1 A B

21. NOR Gate Timing Diagram