Is it possible to implement an XOR gate using only NOR gates?

[kel]

Homework Statement

Part 1 - give the truth table for an XOR port
part 2 - implement the results (from part 1) using only NOR ports

Homework Equations

The Attempt at a Solution

So far I have tried combining NOR gates and have even used single input NOR gates (effectively turning them into NOT gates), but I whatever combination I use I can't see how to make the circuit behave like an XOR gate/port.

I always end up with 2 signals going into a final NOR gate that could be 0&0, 0&1, 1&0 or 1&1, but by definition I thought that an XOR will only have a '1' signal if and only if we have 0&1 OR 1&0 - Which is not the same as a NOR gate. So I can't see how using only NOR gates can work.It would be easy to make the XOR using NOT and AND gates, but I'm not so sure about NOR gates.

Can anyone suggest something? a hint would be appreciated

Thanks
Kel

 

[Päällikkö]

So did you make the truth table?
With NOR (or NAND) you can make all the other gates.

The XOR gate with logical operators: pq + pq, where p and q are the inputs and underline means negation. So what you need to construct from the NORs are the NOT gate (which you said you've done) AND and OR (NOT NOR).

 

[andrevdh]

... The XOR gate with logical operators: pq + pq, ...

I assume the + is the AND operation? What logical operation is implied between p and q and their inverted signals in this equation?

 

[Päällikkö]

I assume the + is the AND operation? What logical operation is implied between p and q and their inverted signals in this equation?

Multiplication is AND, + is OR, I suppose that notation is rather commonly used (and doesn't require LaTeX, which would take a little longer to type).

In other notation:
$$(p \land \lnot q) \lor (\lnot p \land q)$$

 

[kel]

I have the truth table, but could you please explain how to use the NOR gate to make an AND and an OR gate.

This I think is where my difficulty lies, I can't see how to get the inputs to give me the right output so that I can make a XOR gate. But if like you say it's possible to make a NOR work like an AND or OR gate then that would make it fairly easy I should think.

Thanks again
Kel

 

[Mentz114]

A AND B = (not A) NOR ( not B)

 

[andrevdh]

Multiplication is AND, + is OR, I suppose that notation is rather commonly used (and doesn't require LaTeX, which would take a little longer to type).

In other notation:
$$(p \land \lnot q) \lor (\lnot p \land q)$$

So does pq means that p and the inverted signal of q are NORed together?

 

[Päällikkö]

So does pq means that p and the inverted signal of q are NORed together?

No. They're ANDed. AND can be expressed with NORs (see Mentz114's hint).

 

[andrevdh]

Thank you. So I guess his problem is solved, because you get OR by NOTting the NOR (which is constructed by feeding the signal to both legs of a NOR, that is building the NOT function).

 

[Mentz114]

NOT is a single input NOR.

 

[andrevdh]

Ah, so you tie the other input to ground (or does it default to ground [low] if not connected?).

 

[Mentz114]

If we're talking hardware, connect it to [low]. Leaving it floating could have bad effects.

See the first post also where the single input NOR is mentioned.