Implementation of XOR Gate from NOR Gate

Implementation of the XOR gate from the NOR gate is possible because the NOR gate is a Universal gate i.e., it can implement all other gates. The NOR gate gives the output 1 if all the inputs are 0. XOR gate is a logic gate that gives output 1 when several 1s are odd. In this article, we will explore the XOR gate, and NOR gate and implement the XOR gate from the NOR gate.

What is XOR Gate?

XOR gate is a logic gate that results in an output high with an odd number of high inputs. In other words, if the number of 1's is odd in the input then, the output of XOR is 1. XOR gate gives output 1 when all the inputs are different. The XOR is represented as ⊕.

👁 XOR Gate

In the above table if the inputs are different then the output is 1 otherwise 0.

Expression for 2-Input XOR Gate

We get the expression of XOR gate as sum of products of complement of first input with second input and first input with complement of second input. From the above truth table, the expression of XOR gate is:

A ⊕ B = A'B + AB'

What is NOR Gate?

NOR gate is logic gate which results output high when all the inputs are low. In other words, the output is 1 when all the inputs are 0. NOR is a universal gate. It is similar to the complement of OR gate.

2-Input NOR Gate

The NOR gate with 2 input is called 2-input NOR gate. If both the input is low then, NOR gate gives high output. Alternatively, when both input is 0 then output is 1 for NOR gate.

👁 NOR Gate

Expression for 2-Input NOR Gate

From the above table the expression for 2-input NOR gate is:

A NOR B = A'B' = (A + B)'

Logic Diagram for 2-Input NOR Gate

We can get NOR gate in two ways. First, by putting a bubble in the output of OR gate. Second, by putting two bubbles in the input of AND gate.

👁 2-Input NOR Gate

n-Input NOR Gate

The NOR gate with n-input is called n-input NOR gate. When all the inputs are low then output is high. In other words, all inputs are 0 then, output is 1.

Expression for n-Input NOR Gate

If the inputs are A₁, A₂, ..., A_n then,

A₁ NOR A₂ NOR ...NOR A_n = (A₁+ A₂+ ... + A_n)'

Logic Diagram for n-Input NOR Gate

Below is the logic diagram for n-input NOR gate.

👁 n-Input NOR Gate

Implementation of XOR Gate from NOR Gate

Below is the logic diagram for implementation of XOR gate from NOR gate

👁 XOR Gate From NOR Gate

To implement XOR gate from NOR gate we follow below steps

• First, we connect both the inputs in NOR gate resulting in (A + B)'.
• Then, after we have another two NOR gates with the inputs A, (A + B)' and B, (A + B)' respectively with outputs [A + (A + B)']' and [B + (A + B)']'.
• Again, the connect outputs of the above NOR gates to next NOR gate resulting in the output [A'B + AB']'.
• Now, the result is generated in complement form so, we connect the result as the input for last NOR gate which gives the output [AB' + A'B].
• The resultant of the last NOR gate gives us XOR gate.
• So, the number of NOR gates required to implement XOR gate is 5.
  

Conclusion

From the above discussion we can conclude that to implement XOR gate from NOR gate we require 5 NOR gates. In first NOR gate we connect both the inputs A and B. The expression for the XOR gate is given by A'B + AB' whereas the expression for the NOR gate is (A + B)'.