Checking if two expressions are related

Is it possible in mathematica to check if two expressions may be able to be written in terms of the other? The expressions I had in mind were of the form $$\alpha = \frac{1-x}{1-y}\,\,\,\,,\,\,\ \beta = \frac{1+x}{1+y}$$ and was wondering if it's possible to find a relation $$\alpha = f(\beta)?$$

It looks like the answer for the given system is No.

Defining the equations first:

eqns = {a == (1-x)/(1-y), b == (1+x)/(1+y)};

We can use Eliminate to try and suss out a relation which does not depend on x or y.

Eliminate[eqns, {x, y}]

True

However, Eliminate returns True. This means that a and b can be chosen independently of each other, disproving the existence of a relation of the form $$\alpha = f(\beta)$$ immediately. If that's not convincing, however, you can use Reduce to find out the precise conditions for which a solution for a exists:

Reduce[eqns, a, {x, y}]

(b == 1 && a == 1) || (-1 + a) (a - b) (-1 + b) != 0

So if a and b are both 1, or so long as the other expression is not 0, any pairing of a and b should be possible. You can use FindInstance to find a few, if you wish:

FindInstance[eqns /. {a->4, b->2}, {x, y}]
FindInstance[eqns /. {a->5, b->2}, {x, y}]

{{x -> 5, y -> 2}}

{{x -> 13/3, y -> 5/3}}

Solve $$\beta$$ for $$x=x(\beta,y)$$:

S = Solve[β == (1 + x)/(1 + y), x]

{{x -> -1 + β + y β}}

Use this list of solutions to express $$\alpha=\alpha(\beta,y)$$:

α = FullSimplify[(1 - x)/(1 - y) /. S]

{(-2 + β + y β)/(-1 + y)}

Pick those solutions where $$\alpha$$ depends only on $$\beta$$ but not on $$y$$:

Select[α, FreeQ[y]]

{}

No solutions found in this case.

You can use FindInstance to generate a simple counter-example to the hypothesis that a is a function of b

a = (1 - x)/(1 - y);
b = (1 + x)/(1 + y);

FindInstance[{a == 2, b == 1/2}, {x, y}]

(* {{x -> -(1/3), y -> 1/3}} *)

FindInstance[{a != 2, b == 1/2}, {x, y}]

(* {{x -> 1, y -> 3}} *)
• Thanks. This is because if $a = f(b)$ existed then by definition it cannot be one to many.
– CAF
May 4 '19 at 8:58