MaplePrimes - answers and comments on Question, Union of two sets of solution

union

Torre — Wed, 22 Feb 2012 20:06:48 Z

s1 := solve(x^2+ 3*x + 2 >0, {x});
                   
s2 := solve(x^2 - 9>=0, {x});
                
s3 := s1 union s2;

By disjunction

Markiyan Hirnyk — Wed, 22 Feb 2012 20:09:39 Z

Because S1 union S2:= {x in R : x belongs to S1 or x belongs to S2}= {x: x^2+ 3*x + 2 >0 or x^2 - 9>=0 }, we have

> restart; S := solve(x^2+3*x+2 > 0 or x^2-9 >= 0, {x});#union

{x < -2}, {-1 < x}

> `union`(S[1], S[2]);

                              {-1 < x, x < -2}


> T := solve(x^2+3*x+2 > 0 and x^2-9 >= 0, {x});#intersection


> `union`(T[1], T[2]);

dubious

Alex Smith — Thu, 23 Feb 2012 00:59:42 Z

Whatever it is that "union" does in

s1 := solve(x^2+ 3*x + 2 >0, {x});

s2 := solve(x^2 - 9>=0, {x});

s3 := s1 union s2;

is dubious. If you replace "union" with "intersection" then Maple's answer is {}

But the intersections of the solutions sets to the two inequalities is non-empty:

solve({x^2+ 3*x + 2 >0,x^2 - 9>=0});

{x <= -3}, {3 <= x}

 So I don't think union and intersection are doing anything meaningful in this context.

Question is correct

Markiyan Hirnyk — Thu, 23 Feb 2012 01:21:47 Z

As far as I understand it, the question is correct: to find the union of the two sets, which are subsets of R and are formed by the solutions of the given inequalities, with Maple.

PS. The s1 and s2 are not sets, these are expressions, which describe the sets of the solutions:

> whattype(s1);

exprseq

The s1 union s2 command produces the union of these expressions, not the union of the sets and the s1 intersect s2 command produces the intersection of these expressions.

PPS. > whattype({x < -2});

set
> whattype({-1 < x, x < -2});

set

correct

Alex Smith — Thu, 23 Feb 2012 02:49:40 Z

Yes, I agree. Your solution is correct. I was referring to the solution in the post titled union

And pointing out that s1 is a expseq and not a set perfectly explains why using union/intersection in this context is dubious. One might expect that in this context, these operations will produce unions/intersections of solution sets. But they don't.

sets

Alejandro Jakubi — Thu, 23 Feb 2012 09:41:44 Z

Maple sets (and their operations) are "computational", rather than "mathematical". Actually, Maple misses a good representation of sets and their algebra (in particular "abstract" sets in terms of predicates), though it has several mediocre representations for some kind of sets like intervals. And which one solve uses in its output depends on the form of the variable argument. In particular, for a name it uses the property representation, where intervals are represented by RealRange function calls. This is something weird in my opinion, but it has the advantage that implemented operations are more mathematical. E.g. the representation of intersection of intervals is AndProp, and using it the following workaround could be tried for intersecting the different solution intervals and combining the results:

s1 := solve(x^2+ 3*x + 2 >0, x);                 
      s1 := RealRange(-infinity, Open(-2)), RealRange(Open(-1), infinity)
s2 := solve(x^2 - 9>=0, x);
             s2 := RealRange(-infinity, -3), RealRange(3, infinity)
(AndProp@op)~({seq(seq([s1[i],s2[j]],i=1..2),j=1..2)});
         {BottomProp, RealRange(3, infinity), RealRange(-infinity, -3)}

Here, BottomProp is the property equivalent of the empty set.