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)

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.

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)

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)

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)

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

AND Gate Timing Diagram

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

OR Gate Timing Diagram

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

NOT Gate Timing Diagram

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

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

OR Gate Applications Car door open alarm

NOT Gate Applications 1’s Complement

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 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1

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

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

NAND Gate Timing Diagram

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

NOR Gate Timing Diagram