## Introduction :

Logic gates are the digital electronics devices (or) circuits, which accept one (or) more inputs but give only one output.

Why do we need logic gates in the first place? Logic gates are the building blocks for modern digital computation. With the help of logic gates, any kind of logic function can be performed. There are so many different types of logic gates, These are classified depending on their logical operations, ( EX-OR, EX-NOR )among them some are classified as Basic logic gates ( AND, OR, NOT ) and some are classified as Universal logic gates ( NAND, NOR ). Here the EX-NOR logic gate is classified depending upon its logical operation. The electronic circuit that performs this EX-NOR logic function is called the EX-NOR logic gate.

Logic gates are the basic building blocks, for all the modern-day digital electronics devices.

## EX-NOR logic gate :

The logic behind the EX-NOR logic gate is, “If an even number of inputs to a logic gate is logic high, then its output is logic high”. This type of logic gate is known as the EX-NOR logic gate.

EX-NOR logic gate is simply known as “even 1’s detector”. It is a complemented version of the logic EX-OR gate.

EX-NOR means Exclusive-NOR. It is also represented as a logic X-NOR gate.

### What is meant by logic high?

Here logic high means the input voltage of one of the inputs is slightly higher than the input voltage of the other inputs. The most common logic high input voltages are (+5 V, +1 V, +3.3 V etc.,). In digital terms logic high is represented as “1” in positive logic and as “0” in negative logic. In digital terminology “1” is not a mathematical constant one (1), it’s just a representation of logic high state in positive logic, similarly “0” is not a mathematical constant zero (0), it’s just a representation of logic high state in negative logic.

In Boolean algebra terms,

logic high = 1 = high voltage ( positive logic )

logic high = 0 = high voltage ( negative logic )

### What is meant by logic low?

Here logic low means the input voltage of one of the inputs is slightly lower than the input voltage of the other inputs. The most common logic low input voltages are ( 0 V, -5 V, -3.3 V, -1 V etc.,). In digital terms logic low is represented as “0” in positive logic and as “1” in negative logic. In digital terminology “0” is not a mathematical constant zero (0), it’s just a representation of logic low state in positive logic, similarly “1” is not a mathematical constant one (1), it’s just a representation of logic low state in negative logic.

In Boolean algebra terms,

logic low = 0 = low voltage ( positive logic )

logic low = 1 = low voltage ( negative logic )

### Note :

- EX-NOR logic is the same as even 1’s detector.
- If the number of inputs given to the logic EX-NOR gate is odd, then the logic operation of the EX-NOR logic gate is equal to the logic operation of the EX-OR logic gate.
- If the number of inputs given to the logic EX-NOR gate is even, then the logic operation of the EX-NOR logic gate is an inverted version of the logic operation of the EX-OR logic gate.

### EX-NOR operator :

EX-NOR operator is represented as “ ”. Read it as “EX-NOR”. It is represented as a dot (or) binary multiplication operator surrounded by a circle. It acts like odd 1’s detector.

Another way of representing the logic EX-NOR operator is overline above the logic EX-OR operator

EX :

Boolean Expression = , Read it as A “EX-NOR” B

## Classification of logic EX-NOR gates :

EX-NOR gates are classified depending on their number of inputs. The most commonly used EX-NOR gates are

- Two input logic EX-NOR gate.
- Three input logic EX-NOR gate.

### Two input logic EX-NOR gate :

If an EX-NOR logic gate accepts two inputs and provides the desired EX-NOR logic as output then that type of logic gate is called a two-input logic EX-NOR gate.

Boolean logical Expression,

Its logic function is,

#### Positive logic :

“When both of the inputs are the same i.e. Either both of the inputs are logic high (1) or both of the inputs are logic low(0), only then the output is logic high (1)”. Positive logic gates are widely used.

#### Negative logic :

“When both of the inputs are the same i.e. Either both of the inputs are logic high (0) or both of the inputs are logic low(1), only then the output is logic high (0)”

#### Logic Symbol :

#### Switch representation :

If,

n = number of switches (inputs)

m = maximum possible number of output combination

Ex :

number of inputs (n) = 2

Maximum possible number of output combination (m) = 2^{2} = 4

#### Truth table :

##### Positive logic :

If any one input is,

0 = EX-NOR gate acts as an inverter with output being the inverted version of other input.

1 = EX-NOR gate acts as a buffer with the output being the other input.

With this, we can say that,

Positive logic EX-NOR gate is equal to the Negative logic EX-OR gate

##### Negative logic :

If any one input is,

0 = EX-NOR gate acts as a buffer with the output being the other input.

1 = EX-NOR gate acts as an inverter with output being the inverted version of other input.

With this, we can say that,

Negative logic EX-NOR gate is equal to the Positive logic EX-OR gate

##### Unused input :

###### TTL logic :

In TTL logic, if any input is open (or) floating, it will act as “1”

###### ECL logic :

In ECL logic, if any input is open (or) floating, it will act as “0”

### Three input logic EX-NOR gate :

If an EX-NOR logic gate accepts three inputs and provides the desired EX-NOR logic as output then that type of logic gate is called a three-input logic EX-NOR gate.

Boolean logical Expression,

Its logic function is,

#### Positive logic :

“When an even number of the inputs is logic high (1), only then the output is logic high (1)”. Positive logic gates are widely used.

#### Negative logic :

“When an even number of the inputs is logic high (0), only then the output is logic high (0)”.

#### Logic Symbol :

#### Switch representation :

If,

n = number of switches (inputs)

m = maximum possible number output of combination

Ex:

number of inputs (n) = 3

Maximum possible number of output combination (m) = 2^{3} = 8

#### Truth table :

##### Positive logic :

If any one input is,

0 = EX-NOR gate gives EX-OR of the other two inputs as output.

1 = EX-NOR gate gives EX-NOR of the other two inputs as output.

With this, we can say that,

Positive logic EX-NOR gate is equal to the Positive logic EX-OR gate

##### Negative logic :

If any one input is,

0 = EX-NOR gate gives EX-NOR of the other two inputs as output.

1 = EX-NOR gate gives EX-OR of the other two inputs as output.

With this, we can say that,

Negative logic EX-NOR gate is equal to the Negative logic EX-OR gate

##### Unused input :

###### TTL logic :

In TTL logic, if any input is open (or) floating, it will act as “1”

###### ECL logic :

In ECL logic, if any input is open (or) floating, it will act as “0”

## Boolean law :

EX-NOR gate follows all these Boolean laws.

**Commutative law :**

This law states that the logical EX-NOR operation between the two inputs is equivalent to the logical EX-NOR operation performed by changing the order of the same inputs.

i.e.

where, A & B are logic inputs.

**Associative law :**

This law states that if we have three logic inputs, then the logical EX-NOR between the logical EX-NOR of the second and third logic input, to the first logic input, is equal to the logical EX-NOR between the logical EX-NOR of the first and second logic input, to the third logic input.

i.e.

where, A,B & C are logic inputs.

## Statements proof :

### Statement 1 :

If the number of inputs given to the logic EX-NOR gate is even, then the logic operation of the EX-NOR logic gate is an inverted version of the logic operation of the EX-OR logic gate.

### Proof 1 :

### Statement 2 :

If the number of inputs given to the logic EX-NOR gate is odd, then the logic operation of the EX-NOR logic gate is equal to the logic operation of the EX-OR logic gate.

### Proof 2 :

## Conclusion :

Now we all know, what is logic EX-NOR gate and how it operates. Logic gates are the basic building blocks for any kind of digital electronic device/system. Logic EX-NOR gate is also mentioned as logic X-NOR gate. It acts as even 1’s detector. EX-NOR means Exclusive-NOR.