Obsolesence of <> (fwd)

[David Eppstein]

James Logajan (JamesL at Lugoj.Com) wrote:
> : Nicely done! You've worked out how to have your cake and Edith too! Now if I
> : can only think of real world situation where <> does what you want and !=
> : doesn't (since the latter seems to include the former, but note vice versa).
> : Any case where an element of a ordered set can be NOT less than AND NOT
> : greater than another element and still be NOT EQUAL to it? I'm not a
> : mathematician, but do infinities have these properties?

Check out Conway's theory of combinatorial games. Any game has a value; if 
A and B are two game values, then A and B can be equal, A < B, A > B, or 
"fuzzy" (written "||"). These are mutually exclusive possibilities. 
Certain game values are numbers (in Conway's Surreal number system, which 
includes all the familiar real numbers) in which case these comparisons 
are just the same as the usual numeric comparisons, but of course two 
numbers are never fuzzy.

If a game is positive or negative, one of the two players wins (with 
perfect play) regardless of who wins first.  If it is zero, the second 
player wins, and if it is fuzzy with respect to zero, the first player 
wins. Surreal numbers have lots of interesting infinite values, but these 
are numbers (so not fuzzy) -- the fuzzies are near zero in value.

The relevant references are Conway, _On Numbers and Games_, and Berlekamp, 
Conway, and Guy, _Winning Ways_, both of which are I think finally back in 
print.

What this has to do with Python programming I'm not sure...
-- 
David Eppstein       UC Irvine Dept. of Information & Computer Science
eppstein at ics.uci.edu http://www.ics.uci.edu/~eppstein/