# Can (x'y' + xy) be simplified?

I started with (AB' + A'B)' and ended up with (A'B' + AB). Is this all the farther I can go? I feel like this is always going to be true, but I'm not sure how to prove it algebraically.

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$(fg)'=fg'+f'g, (fg)''=f''+2f'g+2fg'+g''$ etc. what are $x$ and $y$? what are you asking? – yoyo Apr 13 '11 at 1:25
sorry, should have used A and B instead of x and y...A and B are booleans. – Marty Apr 13 '11 at 1:33
What do you mean by simpler? Fewer gates (AND/NOT/OR)? Smaller length when viewed as a string? – Aryabhata Apr 13 '11 at 1:37
In a sense, this is the farthest you can go. What you have reached is the equivalence operation, which is the negation of XOR. – Brandon Carter Apr 13 '11 at 2:31
@Brandon: You should probably write that as an answer. – Ben Alpert Apr 13 '11 at 4:29

## 1 Answer

As per request, I am making my comment into an answer.

In a sense, this is the farthest you can go. What you have reached is the equivalence operation, which is the negation of XOR. Equivalence, sometimes denoted XNOR, returns true if the inputs are either both true or both false. See the Wikipedia page for XNOR for more information.

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I have always thought NXOR would be a more logical name than XNOR – Henry Apr 13 '11 at 12:02
thanks, now i see this is an xnor. – Marty Apr 13 '11 at 19:43
@Henry: NXOR would be more logical, but does not have an easy pronunciation like XNOR does. – Brandon Carter Apr 17 '11 at 18:48
"Nex or"? :) – bcat Apr 26 '11 at 4:54
@Henry I've always thought "=" should be used. – Marnix Klooster Jan 16 '15 at 6:53