# How to derive XOR from only NOR gates

• Glavata
In summary, the process for deriving a 2-input XOR gate from only NOR gates is by using the formula (AB)'(A'B')'. This formula represents a circuit where A NOR A is NOR'd with B NOR B, and the result is NOR'd with A NOR B.
Glavata
Hi, Through some searching and guessing with LogiSim I was able to see how to build a XOR gate from only NOR gates. I would like to know what the process for deriving this would look like. Right now I am lost as to how to proceed given that the NOR function is A'B' and XOR is AB' + A'B. How would you get from only NOR gates to building a circuit equivalent to a 2 input XOR.

Thanks!

Welcome to PF.

See if this helps:

http://en.wikipedia.org/wiki/NOR_logic"

Last edited by a moderator:
I think this is what you're looking for:

AB' + A'B
= (0 + AB') + (A'B + 0)
= (AA' + AB') + (A'B + BB')
= A(A' + B') + B(A' + B')
= (A' + B')(A + B)
= (AB)'(A'B')'
= [(AB) + (A'B')]'
= [(A' + B')' + A'B']'
= [(A'A' + B'B')' + A'B']'
= ([(A + A)' + (B + B)']' + A'B')'
= ([(A + A)' + (B + B)']' + (A + B)')'

This formula represents your circuit: A NOR A is NOR'd with B NOR B. The result is NOR'd with A NOR B.

## 1. How do you derive XOR from only NOR gates?

To derive XOR from only NOR gates, you need to first understand the concept of exclusive OR or XOR. XOR is a logical operation that produces a true value only when its two inputs differ. This can be achieved using only NOR gates by connecting two NOR gates together in a specific way. The output of one NOR gate is connected to the input of the other NOR gate, and the inputs of both NOR gates are connected together. This creates a circuit that behaves like an XOR gate, producing a true output only when its two inputs differ.

## 2. What is the advantage of using only NOR gates to derive XOR?

The main advantage of using only NOR gates to derive XOR is that it simplifies the circuit design and reduces the number of gates required. XOR gates typically require three or more gates to be constructed, whereas a single XOR gate can be derived from only two NOR gates. This can save space and reduce the cost of building complex circuits.

## 3. Is it possible to derive XOR from other logic gates besides NOR gates?

Yes, XOR can also be derived from other logic gates such as NAND gates, AND gates, and even a combination of different gates. However, using only NOR gates to derive XOR is preferred as it simplifies the design and reduces the number of gates required.

## 4. Can the XOR function be implemented using only one NOR gate?

No, XOR cannot be implemented using only one NOR gate. This is because a single NOR gate can only produce a true output when both of its inputs are false. To achieve the exclusive OR behavior, at least two NOR gates are required.

## 5. How does the NOR gate's truth table relate to the XOR function?

The NOR gate's truth table is closely related to the XOR function. Both have a true output when their inputs differ, and a false output when their inputs are the same. This is why by connecting two NOR gates together, we can achieve the exclusive OR behavior and derive XOR from only NOR gates.

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