** Definition:-**

**Boolean algebra**is the branch of algebra in which the values of the variables are the truth values

*true*and

*false*, usually denoted 1 and 0, respectively. Instead of elementary algebra, where the values of the variables are numbers and the prime operations are addition and multiplication, the main operations of Boolean algebra are the conjunction (

*and*) denoted as ∧, the disjunction (

*or*) denoted as ∨, and the negation (

*not*) denoted as ¬. It is thus a formalism for describing logical operations, in the same way that elementary algebra describes numerical operations.(source:wikipedia.org)

Or

**Features:**

-> 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’)’ and 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 Ex-OR,Ex-NOR etc) which are used to design digital circuits/boards.

->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 is it used?

**Boolean expression and Boolean function:-**

Boolean expression:-

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.

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.

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’ or A or A or NOTed A. They all have same meaning. Better we can understand it by using Truth table as given below.

From this table we can conclude that this operation gives us ‘1’ (working) output if input or value of variable is ‘1’.

Gates:-

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 two 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 totally 4 input combinations (22=4).

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).

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.

**Electrical circuit:-**

**source:https://www.electronics-tutorials.ws/**

‘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 behaviour 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= A

Its truth table can be made and analysed in following way.

**Electrical circuit:-**

**source:https://www.electronics-tutorials.ws/**

Some other gates:

Derived gate:-

There are some gates which we can get by combining basic gates. 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,

OR

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)

Let’s have a look at truth table and analyse all outputs.

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’

**Electrical circuit:-**

‘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,

or

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.

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..

**Electrical circuit:-**

**Ex-OR’ (Exclusive-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 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=AB

or

C=A’.B+B’.A

Here, is called Ex-OR operation or encircled operation.

Its truth table can be seen as given below.

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 EX-OR 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.

**Electrical circuit:-**

Ex-NOR 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=(AB)’

or

C=A.B+A’.B’

Its truth table can be seen as given below.

Here, we can see that all inputs are giving output by performing ‘Ex-OR’ 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.

**Electrical circuit:-**

**Universal gate:-**

We have two gates namely ‘NAND’ and ‘NOR’ which we can use to design and implement all features of all other gates. Due to having of this property they are called ‘universal gate’. They have capability to mimic other gates and their operations. All logic function we can get by using these two.We can use a number of universal gates to design simple as well complex circuits. So, it makes our work easier.

How to design gate by using ‘NAND’:- We can design gates by using ‘NAND’. Let’s try for all these.

**a) ‘NOT’ by using ‘NAND’ gate:-**

We know that in ‘NOT’, there is one input, and output is in inverted form; we have to get same by using ‘NAND’ gate. Let’s try.

Here, we can see that the input is inverted for each. Now we have to design ‘NAND’ in such a way that it would also give same outputs.

Since in ‘NOt’, there is one input so we take one input as shown above in ‘NAND’. Now let’s analyse its truth table.

Let’s now compare both the outputs; we get same so we have designed the ‘NOT’ gate by using ‘NAND’.

b)‘

**AND’ by using ‘NAND’:-**

We know that in ‘AND’, there are two inputs, and output is in inverted form; we have to get same by using ‘NAND’ gate. Let’s try.

In above table, we have shown all inputs and outputs. Now we have to get same output as ‘AND’ gate has and by using ‘NAND’ gate/s only.

Since there are two inputs in ‘AND’ gate so same has to be taken in ‘NAND’.

Here, we have used two ‘NAND’ gates to obtain ‘AND’ gate. If we compare the values/outputs with ‘AND’ then obviously they are being same so we got it.

C)‘OR’ by using ‘NAND’ gate:-

We know that in ‘OR’, there can be two or more inputs and one output. ‘OR’ gate performs ‘+’ operation. we have to get same by using ‘NAND’ gate. Let’s try with a short look on their truth table.

In above table, we have shown all inputs and outputs. Now we have to get same output as ‘OR’ gate has and by using ‘NAND’ gate/s only.

Since there are two inputs in ‘OR’ gate so same has to be taken in ‘NAND’.

Here, we have used three ‘NAND’ gates to obtain ‘OR’ gate. If we compare the values/outputs with ‘OR’ then obviously they are being same so we got it.

d)

X-OR by using NAND:- We can get ‘X-OR’ gate by using ‘NAND’. Before we make it, let’s call once again all inputs and outputs (truth table).

We know that in ‘X-OR’, there can be two or more inputs and one output. X-‘OR’ gate performs ‘+’ operation for some inputs except one. we have to get same by using ‘NAND’ gate. Let’s try with a short look on their truth table.

In above table, we have shown all inputs and outputs. Now we have to get same output as ‘X-OR’ gate has and by using ‘NAND’ gate/s only.

Since there are two inputs in ‘X-OR’ gate so same has to be taken in ‘NAND’.

Here, we have used four ‘NAND’ gates to obtain ‘X-OR’ gate. If we compare the values/outputs with ‘OR’ then obviously they are being same so we got it

**‘Ex-NOR’ by using ‘NAND’ gate:-**

We can get ‘X-NOR’ gate by using ‘NAND’. Before we make it, let’s call once again all inputs and outputs (truth table).

We know that in ‘X-NOR’, there can be two or more inputs and one output. X-‘NOR’ gate performs ‘+’ operation for some inputs except one and then ‘NOT’ for all. We have to get same by using ‘NAND’ gate. Let’s try with a short look on their truth table.

In above table, we have shown all inputs and outputs. Now we have to get same output as ‘X-NOR’ gate has and by using ‘NAND’ gate/s only.

Since there are two inputs in ‘X-NOR’ gate so same has to be taken in ‘NAND’.

Here, we have used five ‘NAND’ gates (above) to obtain ‘X-NOR’ gate. If we compare the values/outputs with ‘OR’ then obviously they are being same so we got it.

**‘NOR’ universal gate:**- Like ‘NAND’, we have another universal gate called ‘NOR’. We can also implement this to obtain all others.

Let’s get all now.

**‘NOT’ by using ‘NOR’:-**

We know that in ‘NOT’, there is one input, and output is in inverted form; we have to get same by using ‘NOR’ gate. Let’s try.

Here, we can see that the input is inverted for each. Now we have to design NOR’’ in such a way that it would also give same outputs.

Since in ‘NOt’, there is one input so we take one input as shown above in ‘NOR’. Now let’s analyse its truth table.

Let’s now compare both the outputs; we get same so we have designed the ‘NOT’ gate by using ‘NOR’.

‘

**OR’’ by using ‘NOR:**- To get ‘OR, we can follow same procedure as we did above (in case of ‘AND’ by using ‘NAND’. Just we need to change the gate’s symbol. You use following symbol instead of ‘NAND’ and get same output as we have in ‘OR’.

**‘AND’ by using ‘NOR’:-**’ To get ‘AND’, we can follow same procedure as we did above (in case of ‘OR’ by using ‘NAND’. Just we need to change the gate’s symbol. You use following symbol instead of ‘NAND’ and get same output as we have in ‘AND’.

‘

**X-OR’ by using ‘NOR’:-**’ To get ‘X-OR’’, we can follow same procedure as we did above (in case of ‘X-OR’ by using ‘NAND’). Just we need to change the gate’s symbol. You use following symbol instead of ‘NAND’ and get same output as we have in ‘X-OR’. You do not need to change anything else.

‘

**X-NOR’ by using ‘NOR’:-**’ To get ‘X-NOR’’, we can follow same procedure as we did above (in case of ‘X-NOR’ by using ‘NAND’). Just we need to change the gate’s symbol. You use following symbol instead of ‘NAND’ and get same output as we have in ‘X-NOR’. You do not need to change anything else.

Boolean laws:- There are some laws to be understood when we go for design/analysis/simplification of boolean expression. Let’s have a look at all of them.

a) Identity law:- If ‘A’ is a Binary variable then it states:

a.1) A+0=A a.2)A.1=A

proof:-Taking L.H.S.,

=RHS

Hence, proved.

proof a.2)

=RHS

Hence, proved.

b)Complement law:- If ‘A’ is a Binary variable then it states:

a.1) A+A’=1 a.2)A.A’=0

proof:-Taking L.H.S.,

=RHS

Hence, proved.

proof a.2)

=RHS

Hence, proved.

C)Idempotent law:- If ‘A’ is a Binary variable then it states:

a.1) A+A=A a.2)A.A=A

proof (a.1):-Taking L.H.S.,

=RHS

Hence, proved.

proof (a.2):-Taking L.H.S.,

=RHS

Hence, proved.

d)Boundedness law:- If ‘A’ is a Binary variable then it states:

a.1) A+1=1 a.2)A.0=0

proof (a.1):-Taking L.H.S.,

=RHS

Hence, proved.

proof (a.2):-Taking L.H.S.,

=RHS

Hence, proved.

e) Absorption law:- If ‘A’ and ‘B’ are Binary variables then it states:

a.1) A+(A.B)=A a.2)A.(A+B)=A

proof (a.1):-Taking L.H.S.,

=RHS

Hence, proved.

proof (a.2):-Taking L.H.S.,

=RHS

Hence, proved.

f)Commutative law:-If ‘A’ and ‘B’ are Binary variables then it states:

For’OR’ operation and for ‘AND’ operation respectively,

a.1) A+B=B+A a.2)A.B=B.A

proof (a.1):-Taking L.H.S.,

Hence, LHS=RHS, proved.

Proof(a.2):- Taking LHS and RHS,

=RHS

Hence, proved.

G) Associative law:- If ‘A’ ,’B’ and ‘C’ are three binary variables then it says that,

For ‘OR’ operation, For ‘AND’ operation,

G.1) A+(B+C)=(A+B)+C G.2) A.(B.C)=(A.B).C

Proof G.1) Taking LHS and RHS together,

Here, LHS=RHS so, law verified.

Proof G.2) Taking LHS and RHS together in different column,

Here, LHS=RHS so, law verified.

H) Distributive law:- If ‘A’ ,’B’ and ‘C’ are three binary variables then it says that,

G.1) A.(B+C)=A.B+A.C G.2) A+(B.C)=(A+B).(A+C)

Proof G.1) Taking LHS and RHS together,

Here, LHS=RHS so, law verified.

Proof G.2) Taking LHS and RHS together in different column,

Here, LHS=RHS so, law verified.

I)Involution law:- IF ‘A’ is a binary variable then it states,

(A’)’=A

proof:-

Taking LHS in truth table,

=RHS

Hence, verified.

J) De-morgan’s law:- If ‘A’ and ‘B’ are two binary variables then it state that:

J.1) (A+B)’=A’.B’ J.2) (A.B)’=A’+B’

proof J.1):- To prove, let’s take truth table with both the sides.

LHS=RHS

proved.

proof J.2):- To prove it, let’s take truth table with both sides.

LHS=RHS so, law proved.

Venn diagram:- It’s a diagram which we use to represent Boolean expression/gates/operations.

Or

A visual representation of all Boolean expression/gates/operations.

Or

A graphical representation of all Boolean expression/gates/operations is Venn diagram.

This diagram gives us a clear idea about operations going on there in the form of coloured circles. We can also understand some other parts which are left uncoloured.

How to use Venn diagram:- We use circles, rectangular boxes, shaded regions etc. We consider,

->each circle as an input (with value ‘1’ or its combinations)

->outer part or space as ‘0’ or its related combinations ‘00’.

->common part for both as ‘11’ or ‘111’.

When we draw venn diagram, we ‘colour/shade’ that ‘part/region’ which is giving us output ‘1’. All other parts are left uncoloured.

To be more clear, let’s have a look at each of gates.

1.)’AND’:- It venn diagram can be drawn as ,

The shaded part is A.B because this part gives output ‘1’ (from truth table).

B)’OR’ gate:-Its venn diagram can be shown below.

The shaded part is A+B because this shaded part gives us output ‘1’. Loot at truth table.

C) ‘NOT’ gate:- Its venn diagram can be represented as,

The shaded part is A’ because it is giving us output ‘1’. once look at truth table.

D) ‘NAND’ gate:- Its venn diagram can be shown below.

The shaded part is (A.B)’ because of output generated by it. Look at truth table.

E)NOR gate:- Its venn diagram can be shown below.

The shaded part is (A+B)’. We have shaded that part which is giving us output ‘1’. Look at following.

F) Ex-OR gate:- Its venn diagram is shown below.

The shaded part is (A X-OR B). The shaded parts are giving us output ‘1’. Look at following truth table.

G) Ex-NOR gate:- Its diagram is shown below.

**Boolean algebraic expression simplifications:-**

**.**

**xyz+y+xy`**

solution:-

**2. (AB+C)`+(AB)`+B`**

solution:-

**3. (ABC+AC+AW)+1**

**=1**[Boundedness law]

**4.pqr+qr+r'**

**x'z+y+x'.1+(y'x)'**

**Given,**

**x'z+y+x'.1+(y'x)'**

**=x'(z+1)+y+(y')'+x' [**De-morgans law+Distributive law]

k xa GEDA

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