Implementation of XNOR Gate using NOR Gate

The NOR gate is a type of universal logic gate, hence we can use only NOR gates to realize the XNOR logic function.

Designing an XNOR Gate using NOR Gate

To design an XNOR gate using NOR gate, we first derive the XNOR logic function in terms of NOR logic as follows.

The output of XNOR gate is given by,

$$\mathrm{Y \: = \: AB \: + \: \bar{A} \: \bar{B}}$$

To implement an XNOR gate logic using NOR gate, we require a minimum of 4 NOR gates. The circuit diagram of XNOR gate from NOR gate is depicted in the following figure:

In this circuit, the output is

$$\mathrm{Y \: = \: \overline{\overline{A \: + \: \overline{(A \: + \: B)}} \: + \: \overline{B \: + \: \overline{(A \: + \: B)}}}}$$

$$\mathrm{Y \: = \: \overline{\overline{A \: + \: \overline{(A \: + \: B)}}} \: \cdot \: \overline{ \overline{B \: + \: \overline{(A \: + \: B)}}}}$$

$$\mathrm{Y \: = \: (A \: + \: \overline{(A \: + \: B)}) \: \cdot \: (B \: + \: \overline{(A \: + \: B)})}$$

$$\mathrm{Y \: = \: (A \: + \: (\bar{A} \: \cdot \: \bar{B})) \: (B \: + \: (\bar{A} \: \cdot \: \bar{B}))}$$

$$\mathrm{Y \: = \: (A \: + \: \bar{A}) \: (A \: + \: \bar{B}) \: (\bar{A} \: + \: B) \: (B \: + \: \bar{B})}$$

$$\mathrm{Y \: = \: (A \: + \: \bar{B}) \: (\bar{A} \: + \: B)}$$

$$\mathrm{Y \: = \: A\cdot\bar{A} \: + \: \bar{A}\cdot\bar{B} \: + \: A\cdot B \: + \: B\cdot\bar{B}}$$

$$\mathrm{\therefore \: Y \: = \: A\cdot B \: + \: \bar{A}\cdot\bar{B}}$$

This is the desired output of the XNOR gate. Thus, the above shown NOR logic circuit performs the XNOR operation.