Designing an XOR Gate from NOR Gates: Truth Tables and Boolean Algebra

[AdkinsJr]

Homework Statement

Use truth tables for a NOR gate to design an XOR gate.

I'm trying to understand how to design simple gates using boolean algebra and I'm getting nowhere. An example I'm working on is a XOR gate built from NOR gates, I already know how to wire it to a breadboard and it works, but I don't understand how to derive the configuration (I attached a diagram). Topics that come up are boolean algebra and Karnaugh maps which are supposed to help, but I can't seem to get instructors or other students to explain it to me how to use this kind of math to actually design a logic circuit.

I attached the XOR diagram as well as truth tables, Karnaugh Map, and the corresponding representation of the function in boolean algebra.

The XOR gate has 5 NOR gates wired as shown of course I just looked it up, but there is no way I would ever be able to come up with that configuration myself. Any tips or advice? Where do you even start?

 

[phinds]

For something as simple as an XOR gate, just think about what it IS. It represents a mildly complex Boolean statement, which in words is "one or the other but not both" or more in gate-like terms, "(A and not B) or (B and not A)". Now break that down. Take the first term. A and not B. Since you can trivially simply wire a NOR gate to give you an inverse, that part is taken care of and basically need to make an AND gate out of NOR gates. Can you figure out how to do that on your own? Then take the other half and do the same kind of thing.

Now, that brute force approach won't always give you the SIMPLEST circuit, but it will give you one that works and learning how to get that far is the first step. As with so many kinds of problems, just break it down and do it little by little.

 

[NascentOxygen]

phinds has helpfully reminded you of the XOR function: A⊕B = A• ̚ B + ̚ A•B

But De Morgan shows how to implement the AND function using the OR function: A•B = ̚ ( ̚ A + ̚ B)

So you can replace each of the ANDs in the top line with its equivalent NOR

Any inverters needed can be formed also using any inverting gate, e.g., a NOR

 

[AdkinsJr]

Before assembling the AND gate from NOR gates let me see if I can make the XOR gate more intuitively..

I can take the first term and get the "not B" by wiring the inputs of the NOR gate together, so the truth table just becomes an inverter for that chunk. ...then I have to "AND" the output of the inverter with "A." So now I have AB*...I can do the same thing then to get the A*B term... and then "or" the outputs AB* and A*B to finally get my function XOR = AB*+A*B...

I think the diagram I attached of this is correct, I used NOR, AND and OR gates...

 

[NascentOxygen]

Yes, that's okay as far as it goes.