Implementation of XNOR Gate from AOI Gate

Digital electronics revolve around storing data in bits of 1s and 0s. Not only is digital electronics an efficient way of storing data but it also makes it easier to encrypt and secure our data. It is awesome how we can use different logic gates to implement some functions and represent these equations.

In this article, we will see how we can represent an XNOR gate using AOI (AND, OR, NOT) gates. Firstly, we will individually understand these gates and then we will implement them.

What is an XNOR Gate?

An XNOR gate is a gate that stands for a combination of "Exclusively-OR" and "NOT" Gate. It gives one input when fed with two or more than two inputs. It is important to note that the output of the XNOR gate is similar to a "NOR" gate.

It becomes low when the any of the is "1" and it becomes high only when both the inputs are same (either '0' or '1'). Therefore it functions in a way that it is high only when either of its inputs is "1" but not all are "1".

Truth Table of XNOR Gate

Here is a truth table that helps us analyze what will be the output of XNOR for different A and B fed to it

Let us see the XNOR gate for two inputs :

The expression for the XNOR gate is given below :

output, Y= (A ⊕ B )'

Where ⊕ and ' represents XNOR gate

What is an AND Gate?

An AND gate is a basic gate that is popularly known as an all-or-nothing gate. It gives one input when fed with two or more than two inputs.

It is important to note that the output of the AND gate becomes low even if any one of the inputs is low and it becomes otherwise. Therefore it functions in a way that it is high only when all its inputs are high or '1'.

Truth Table of AND Gate

Here, We have drawn a Truth table that helps us analyze what will be the output of AND for different A and B fed to it.

Let us see the AND gate for two inputs :

The expression for the AND gate is given below :

output, Y=A.B

where, . represents AND gate

What is an OR Gate?

An OR gate is a basic gate that implements logical disjunction. It gives one input when fed with two or more than two inputs.

It is important to note that the output of the OR gate becomes low only if all the inputs are low and it is high otherwise. Therefore it functions in a way that it is high even when either of its inputs is high or '1'. It works like two switches in parallel meaning when either of the switches is on the light is on.

Truth Table of OR Gate

Here, We have drawn a Truth table that helps us analyze what will be the output of OR for different A and B fed to it

Let us see the OR gate for two inputs :

The expression for the XOR gate is given below :

output, Y=A+B

where, + is OR gate

What is a NOT Gate?

A NOT gate is a basic gate that implements logical complement. It gives one input when fed with one or more than one inputs.

It is important to note that the output of the NOT gate becomes low when input is high and it is high when input is low. Therefore it functions as an invertor and gives '1'. when input is '0' and vice-versa.

Truth Table of NOT Gate

Here, We have drawn a Truth table that helps us analyze what will be the output of NOT for different fed to it

Let us see the NOT gate for an input :

The expression for the NOT gate is given below :

output, Y= A'

where ' represents NOT gate

Implementation of XNOR gate from AOI (AND, OR, NOT) gates

Based on our knowledge of logic gates, we can say that it will require a minimum of 2 AND ,2 NOT and 1 OR gates that will implement the XNOR logic. This means that more than the given number of gates can also implement XNOR logic.

Let us see the implementation below.

The leftmost AND gate is fed with two inputs, the output will be

Y₁= (A.B)

The input fed to the upper NOT is A, therefore the output will be

Y₂= (A)'

The input fed to the lower NOT is B, therefore the output will be

Y₃= (B)'

The output of these two NOT gates i.e. Y₃ and Y₂ will be the input of the second AND gate. Therefore, the output of the 2^nd AND gate is

Y₄= (Y₂.Y₃)' Y₄= (A' . B')

The last OR gate will be fed with Y₄ and Y₁ as the input. Therefore, the final output will be

Y₅=Y₄ + Y₁ Y₅= (A'B'+ BA) Y₅= (A ⊕ B )'

We see that the final output represents XNOR logic. Since the final output is the output of the XNOR gate means our realization of the circuit is correct and we build the correct circuit.

Conclusion

Here we saw how we can use the basic gates called the 'AOI' i.e. AND, OR, NOT gates to implement the 'XNOR' gate. This is important as it helps us analyze the working behind basic gates and we can understand how we can implement it using other gates. This is a basic-level implementation of gates and there can be many more complex realizations using other basic gates. It is advised that readers go through all other realizations to understand the basic functioning behind other important gates. Some Frequently Asked Questions have been answered that cover some other basic concepts. Readers can refer to these questions as well.