Re: [boost] Interval Library and comparison operators

From: Joel Young (jdy_at_[hidden])
Date: 2002-09-06 10:42:46

Hi Doug,

From: Douglas Gregor <gregod_at_[hidden]>
> where a <= x <= c, b <= y <= d
is there a typo in this?  Should it be:
  where a <= x <= b, c <= y <= d

assuming there is a typo,

Just to understand this I quickly came up with the following
three-valued logic partionings of the interval-interval space for:

[a,b] RelOp [c,d] iff 
  Conjunction of x RelOp y for all (x, y) 
    where a <= x <= b, c <= y <= d

Where Conjunction is three-valued conjunction.

       true              false           indeterminate
------------------------------------------------------
<  ==  {<}               {>,mi}          A - {<,>,mi}            
<= ==  {<,m}             {>}             A - {<,>,m} 
=  ==  {}                {<,>}           A - {<,>} 
!= ==  {<,>}             {}              A - {<,>} 
>= ==  {>,mi}            {<}             A - {<,>,mi}
>  ==  {>}               {<,m}           A - {<,>,m}

where A = {<,>,m, mi, o, oi, s, si, f, fi, d, di, =}

Except for (IMHO verboten) point-intervals, true(=) is empty?

Are the existential versions of interest?: 

[a,b] RelOp [c,d] iff 
  Conjunction of 
    (for all x in [a,b] there exists a y in [c,d] such that x RelOp y) 

or

[a,b] RelOp [c,d] iff 
  Conjunction of 
    (for all y in [c,d] there exists a x in [a,b] such that x RelOp y) 

or

[a,b] RelOp [c,d] iff 
  Conjunction of 
    (there exists a x in [a,b] and a y in [c,d]) such that x RelOp y) 

> Is that clearer?

I think so, but if I screwed up above, it must not be. :-)

Joel