# The mathematical proof of a monotonic utility transformation does not restrict the use of strictly decreasing monotonic functions. Why bar them?

I understand from an intuitive sense that decreasing monotonic transformations will skew the choices and ordinality.

But mathematically the $$F'(U(x,y))$$ just cancels out each other out in numerator and denominator when we take the MRS and thus gives us the same MRS as our old utility function. But this cancelling out can also happen with decreasing monotonic transformations. Then why bar them from use ?

The single most important feature that we care about a utility function is that if a bundle $$X$$ is preferred to a bundle $$Y$$ then $$U(X)>U(Y)$$ (with weak inequalities whenever appropriate). This is how we define a utility function. You can immediately see that if a function has that property, any monotonic increasing transformation of it will also have that property. However, if the transformation is decreasing, call it $$V(X)=f(U(X))$$ with $$f'<0$$, then $$V(X) i.e. now $$X$$ gives less utility. Therefore, we see that $$V$$ cannot represent the same preferences.