Equivalences. Re: Checking equivilence of equalities

On 02/08/2012 04:09 PM, Richard Fateman wrote:
> e1: A(x)=B(x)     and e2: C(x)=D(x)   are "equivalent"  if
>
> for all values of x ,    A(x)=B(x)  implies C(x)=D(x)   AND
> C(x)=B(x)  implies A(x)=B(x).
>
> As Stavros pointed out, if you solve(e1,x)   and solve(e2,x)  and the 
> solutions are the same,
> then you are done, and Maxima has done the job.
>
> If you want another way of saying this,  you could ask
>
> zeroequivalence of (C-D) subject to the relation   A-B=0   and
>
> zeroequivalence of (A-B) subject to the relation C-D=0.
>
> A potentially useful goal that may have more general traction is  
> finding an algorithm
> where you could reduce both  e1 and e2 to
> a simpler form and then, even though you could not solve that, you 
> could show they
> are the same.
>
> for example,
> e1:  f(x)+4= g(x)+6
> e2:  f(x)-2 =g(x)
>
> could both be reduced to (say)
>
>  f(x)=g(x)+2 

In some sense in this second case you are also solving the equations and 
comparing them:

(%i1) solve(f+4=g+6, f);
(%o1) [f = g + 2]
(%i2) solve(f-2=g, f);
(%o2) [f = g + 2]

And again, this second example worked because we were able to find the 
solution.

Cheers,
Jaime