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Boolean Algebra

                                                            Boolean Algebra
Meaning and features:­
­> It is a logical mathematics developed by mathematician George Boole.
­> It is used in the world of digital appliances/world.
­>It only uses two variables/values ‘0’ and ‘1’. Here, ‘0’ is called off/false and ‘1’ is called            on/true.
­>It does not use /have exponent values. Like, x*x=x, not x2
­>It does not use/have coefficient values with variables. Like, x+x=x ,not 2x.
­>This mathematics uses operations ‘addition(‘+’ called ‘OR’)’ ,multiplication (called ‘.’ AND and NOT (‘­’,complement or    ¬).
­>It follows all necessary laws Like commutative,associative,distributive and many others.
­> It uses concept of ‘Gates’/’switches’ (AND,OR, NOT X­OR,X­NOR etc) which are used                   to design digital circuits/boards.

 Conventional (Ordinary)algebra:­
               In our daily life we use mathematics for simple/complex calculations.This is called conventional algebra. It can be with following features.
­>It’s a general purpose mathematics used in daily life. It uses/based on oldest number called decimal no. system with digits 0,1,2,3,4,5,6,7,8,9.
­>We can say; it is not used in digital world.
­>It can use many variables with different values. There is no meaning like off or on,true/false in this.
­>Unlike Boolean, x+x=2x and x.x=x2;
 means to say that there is power and coefficient.
­>It uses many operations like addition,subtraction,multiplication,division (unlike Boolean).
­>It does not follow all necessary rules/laws as like in Boolean.
­> This mathematics does not know/say about,what is gate, where it is used?

 Boolean expression and Boolean function:­

 1. Boolean expression:­
Let, A and B are binary variables. It means that they work with or can have only two values 0 and 1 only. Then, an expression formed by combining these variables and operations (AND, OR ,NOT) is called boolean expression.
Examples:
a) A+B, here, A and B are variables and ‘+’ is an ‘OR’ operation.
b) (A.B)+C
c) (A+B.C)
d)A’+B.C
etc.

 Boolean Function:­ It’s a function which is formed by combining binary variables and operations.
                                                                     Or
A function defined by f:A­->B for binary variables ‘A’ and ‘B’ for all inputs of ‘A’ with related outputs of  ‘B’. It is defined in same as we do in normal set theory but variables we take are binary.
we can have example,
f(A,B)=(A+B)+A
f(A,B)=A.B+B’
  etc.
Here, we can see variables used on left and expression on right side.

 Boolean operations:­
Boolean algebra does or has only three types of operations (calculations) namely ‘AND’, ‘OR’ and ‘NOT’. Let’s know  in detail about them.
‘AND’:­
A type of operation which means to go for product or multiplication for given binary variables/inputs. After getting product, we get related output in the term of ‘0’ or ‘1’. For two binary variables A and B, we symbolize ‘AND’ operation by writing A.B or A AND B or A ANDed B. They all have same meaning. Better we can understand it by using 'Truth table' as given below.
Input ‘A’
Input ‘B’
Output  (A.B)
0 (F)
0 (F)
0 (F)
0 (F)
1 (T)
0 (F)
1(T)
0 (F)
0 (F)
1(T)
1(T)
1(T)
From this table we can conclude that this operation gives us ‘1’ (working) output. But in other cases it does not give us if any one value of variable is ‘0’.
‘OR’’:­
A type of operation which means to go for sum or addition for given binary variables/inputs. After    getting sum, we get related output in the term of ‘0’ or ‘1’. For two binary variables A and B, we symbolize ‘OR’  operation by writing A+B or A OR B or A ORed B. They all have same meaning. Better we can understand it by using 'Truth table' as given below.
Input ‘A’
Input ‘B’
Output  (A+B)
0 (F)
0 (F)
0 (F)
0 (F)
1 (T)
1 (T)
1(T)
0 (F)
1 (T)
1(T)
1(T)
1(T)
From this table we can conclude that this operation gives us ‘1’ (working) output in three cases when anyone variable carries value ’1’. But in first case it does not give us if both the values of variables are ‘0’.
‘NOT’’:­
A type of operation which means to go for complement or just reverse of input for given binary    variable/input. This operation uses only one variable/input;unlike previous operations. After reversing the input we get related output in the term of ‘0’ or ‘1’. For binary variable A, we symbolize ‘NOT’ operation by writing A
                            as  A' or ¬ A or NOTed A. They all have same meaning. Better we can understand it by using Truth table as given below.
A
Output A’
0(F)
1(T)
1(T)
0(F)
From this table we can conclude that this operation gives us ‘1’ (working) output if input or value of variable is ‘1’.
Gates (Logic gate):­
It is a basic or major part of digital world. It is an electronic switch or electronic component with 2 or more inputs/signals and one output. All inputs and output work in two states namely ‘0’ and ‘1’. As many inputs are there, there can be many input combinations. Besides, It does not use any other signals. Almost all gates perform some operation. Gates play vital role while designing circuits.
Types:­
Gates are of three types.
a) Basic gates (primary/fundamental/main):­ There are basically three types of gate. They are
‘AND’, ‘OR’ and ‘NOT’ .Let’s know in detail.
a.1) ‘AND’ gate:­
It is an electronic switch or electronic component with 2 or more inputs/signals and one output. All inputs and output work in two states namely ‘0’ and ‘1’. As many inputs are there, there can be many input combinations. This gate performs ‘AND’ (.) operations for all inputs.
Its graphical symbol is,
Here, ‘A’ and ‘B’ are binary inputs/signals and ‘C’ is binary output. Its Boolean expression is,
C=A.B or A AND B
Now, let’s analyse its working principle with the help of truth table. Since we are taking two inputs so there would be  4 input combinations (2 2=4).

Input ‘A’

Input ‘B’

Output C=A.B
0(F)
0(F)
0(F)
0(F)
1(T)
0(F)
1(T)
0(F)
0(F)
1(T)
1(T)
1(T)

So, from this truth table we can conclude that a circuit using two switches (A and B)with an object (bulb) would work (bulb glowing) in only one case, that is last; in last case both the switches are ‘on’ and output is on so. And in other cases, it does not work because of operation and behaviour.
a.1) ‘OR’ gate:­
It is an electronic switch or electronic component with 2 or more inputs/signals and one output.            All inputs and output work in two states namely ‘0’ and ‘1’. As many inputs are there, there can be many input  combinations. This gate performs ‘OR’ (+) operations for all inputs.
Its graphical symbol is,
Here, ‘A’ and ‘B’ are binary inputs/signals and ‘C’ is binary output. Its Boolean expression is,
C=A+B or A OR B
```Now let’s analyse its working principle with the help of truth table. Since we are taking two inputs so there would be   totally 4 input combinations (22=4).

Input ‘A’

Input ‘B’

Output C=A+B
0(F)
0(F)
0(F)
0(F)
1(T)
1(T)
1(T)
0(F)
1(T)
1(T)
1(T)
1(T)

So, from this truth table we can conclude that a circuit using two switches (A and B)with an object (bulb) would work  (bulb glowing) in three cases, that are 2nd 3rd and last; in these cases both the switches are ‘on’ and output is on so. And in other case (both are off), it does not work because of operation and behaviour.
‘NOT’ gate:­
It is an electronic switch or electronic component with 1 input/signal and one output. All   input and output works in two states namely ‘0’ and ‘1’. As only one input can be there, there can be 2 input       combinations. This gate performs ‘NOT’ (complement) operations for all inputs.Its behavior is different than that of other gates because it inverts the incoming signal. So, it is also called ‘Inverter’.
Its graphical symbol is,


Here, ‘A’ is a binary variable or an input and ‘B’ is output.
Its Boolean expression can be written as,
B=A’
or
B=  ¬A
or
B= 
Its truth table can be made and analysed in following way.
Input ‘A’ Output B=A’ or    ¬A

Input ‘A’

Output B=A’ or ¬A
0(F)
1(T)
1(T)
0(F)

Some other gates:­
Derived gate:­ There are some gates which we can get by combining basic gates. These gates have all features of basic gates from where it is derived. NAND,NOR, X-OR (Ex-OR),X-NOR(Ex-NOR).  Let’s look at them in detail.
a)’NAND’ gate:­ A type of electronic switch which is formed by combining two gates namely ‘AND’ and  ‘NOT’. Like others, It also can have 2 or more inputs and only one output. All inputs and outputs work in two states (‘0’ and ‘1’) only. As many inputs are there, there can be many input combinations. This gate performs ‘AND’ (.) and then  ‘NOT’ (complement) operations for all inputs.
Its graphical symbol is,

 
                            
          Here, ‘A’ and ‘B’ are binary inputs/signals and ‘C’ is an Binary output
Its Boolean expression can be written as,
C=(A.B)’
or
C= ¬(A.B)
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Let’s have a look at truth table and analyse all outputs.

Input ‘A’

Input ‘B’

Output C=(A.B)’
0(F)
0(F)
1(T)
0(F)
1(T)
1(T)
1(T)
0(F)
1(T)
1(T)
1(T)
0(F)

So, from this truth table we can conclude that a circuit using two switches (A and B)with an object (bulb) would work                                        
(bulb glowing) in three cases, that are 1st, 2nd 3rd; in these cases both the switches are ‘on’ and output is on so. And in   other case (both are ON), it does not work because of operation and behaviour.
                                         This gate has special property and that is, it is called ‘Universal gate’.
‘NOR’ gate:­ A type of electronic switch which is formed by combining two gates namely ‘OR’ and ‘NOT’.    Like others, It also can have 2 or more inputs and only one output. All inputs and outputs work in two states (‘0’ and ‘1’)   only. As many inputs are there, there can be many input combinations. This gate performs first ‘OR’ (+) and then ‘NOT’  (complement) operations for all inputs.
Its graphical symbol is,
                               
Here, ‘A’ and ‘B’ are binary inputs/signals and ‘C’ is a Binary output.
Its Boolean expression can be written as,
C=(A+B)’
or
C= ¬(A+B)
Let’s have a look at truth table and analyse all outputs.

Input ‘A’

Input ‘B’

Output C=(A+B)’
0(F)
0(F)
1(T)
0(F)
1(T)
0(F)
1(T)
0(F)
0(F)
1(T)
1(T)
0(F)

So, from this truth table we can conclude that a circuit using two switches (A and B)with an object (bulb) would work  (bulb glowing) in one case only, that is 1st case; in this case both the switches are ‘off’ and output is on so. And in other     cases (both are ON or one is ON and other is OFF ), it does not work because of operation and behaviour of ‘NOT’   gate..
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‘X­OR’ (Exclusive­OR)gate:­
It is an electronic switch or electronic component with 2 or more inputs/signals and one output. All
inputs and output work in two states namely ‘0’ and ‘1’. As many inputs are there, there can be many input combinations.  This gate performs ‘OR’ (+) operations for some inputs but exclusive OR operation for others. We can understand this  from given truth table. Its operation is indicated by  ⊕
Its graphical symbol is,

Here, A and B are binary inputs/variables and ‘C’ is output.
Its Boolean expression is,
C=A ⊕B
or
C=A’.B+B’.A
Here, ⊕ is called Ex­OR operation or encircled operation.
Its truth table can be seen as given below.

Input ‘A’

Input ‘B’

Output C=A⊕B
0(F)
0(F)
0(F)
0(F)
1(T)
1(T)
1(T)
0(F)
1(T)

1(T)

1(T)

0(F)

Here, we can see that all inputs are giving output by performing ‘OR’ operation. But in last case, It is giving ‘0’unlike OR gate output. So, for last case we have to exclude as compared to ‘OR’ gate. So why it is called EX­OR gate. It means a circuit having this gate works in two cases only when any one input is ’ON’. And in  other cases, it (bulb) does not work.
XNOR gate:­
It is an electronic switch or electronic component with 2 or more inputs/signals and one output.            All inputs and output work in two states namely ‘0’ and ‘1’. As many inputs are there, there can be many input  combinations. This gate performs ‘OR’ (+) operations for some inputs and then ‘NOT’ operation. We can understand this    from given truth table. Its operation is indicated by complement of    encircled OR operation. ⊕
Its graphical symbol is,

Here, A and ‘B’ are binary inputs and ‘C’ is binary output.
Its Boolean expression can be written as,
C=(A ⊕B)’
or
C=A.B+A’.B’
Its truth table can be seen as given below.

Input ‘A’

Input ‘B’

Output C=(A⊕B)’ or C=A.B+A’.B’
0(F)
0(F)
1(T)
0(F)
1(T)
0(F)
1(T)
0(F)
0(F)

1(T)

1(T)

1(T)

Here, we can see that all inputs are giving output by performing ‘X­OR’ and then ‘NOT’ operation.
It means a circuit having this gate works in two cases only when both of inputs are either ‘0’ or ‘1’. And in other cases, it (bulb) does not work.



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