wheresrhys

I have, as promised, found a better form for XOR:

• XOR(x,y) = x + y -2xy

As I sit here chomping on a chicken wing, I can’t help but feel a touch of disappointment alongside the inevitable satisfaction at completing my mission. The previous solution – XOR(x.y) = (1-xy)(x + y) – was, I feel, more elegabt; the fact we were dealing with x/y symmetry, and that x and y could take only the values 1 and 0, seemed to almost leap out.

Talking of symmetry – I’ve just realised that I didn’t cover x => y, the only asymmetric elemental* logical operator.

So here, deduced by trial and error is the formula:

• =>(x,y) = xy + 1 – x

I lied though. Because, of course, => is not as elemental as one would hope, so:

x => y <–> NOT (x AND NOT(y)) = NOT(x AND (1-y)) = NOT(x(1-y)) = 1-(x-xy) = xy +1 -x

Thus demonstrating the usefulness of being able to represent logical operators analytically.

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Tags: =>, implication, logic, polynomial

This entry was posted on Monday, March 23rd, 2009 at 12:42 am and is filed under Computers and all that, Maths, Philosophy. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.