How can I express the NOT in terms of AND, XOR, XNOR

Question

I need to figure out how to express the NOT operator in terms of the operators AND, XOR, XNOR.

I need to show that this set is functionally complete, which I'm trying to do by showing that I can represent NOT and OR, a known complete set. But I have no idea where to begin when it comes to representing a unary operator (NOT) using binary operators.

EDIT: I figured it out. NOT A=(A XNOR A) XOR A

What the **** is the question? Do we have XNOR or NOR or what??? — Rob Arthan, Feb 07 '16 at 21:03
It was corrected in the post: we have XOR, AND, XNOR. My question has been answered though. Thanks! — zulfe, Feb 07 '16 at 21:18
Good for you. What is left for posterity is a mess, because the accepted answer doesn't answer the question. Posterity will have to read the comments and edits carefully to figure out what the answer actually is. — Rob Arthan, Feb 07 '16 at 22:23
I am new here and didn't know what to do when after realizing I had made an error. Sorry for the confusion. — zulfe, Feb 08 '16 at 01:55

score 3 · Answer 1 · answered Feb 07 '16 at 19:36

3

Hint: $x\ \textrm{OR}\ x = x$.

Hint 2: $x\ \textrm{NOR}\ y = \textrm{NOT}\left(x\ \textrm{OR}\ y\right)$.

answered Feb 07 '16 at 19:36

fgp

21,438

Sorry I actually made a mistake typing out the question. I meant to say XOR instead of NOR. I have edited the question to reflect this – zulfe Feb 07 '16 at 19:54
1

@zulfe Well, then the Hint is that $x\ \textrm{XNOR}\ x = 1$ and that $x\ \textrm{XOR}\ 1 = \textrm{NOT}\ x$. – fgp Feb 07 '16 at 20:33

score 1 · Accepted Answer · answered Feb 07 '16 at 20:26

1

Since you have figured out the $\sf NOT$: to emulate $\sf OR$, what about $$ \textsf{XOR}(\textsf{XOR}(x,y),\textsf{AND}(x,y)) $$ ?

answered Feb 07 '16 at 20:26

Clement C.

68,437

This works great, thank you. I was trying to figure it out using my NOT construction but this is definitely more clear. – zulfe Feb 07 '16 at 20:36

How can I express the NOT in terms of AND, XOR, XNOR

2 Answers2