Does the probability of inequality $X<Y$ change if you transform both variables?

Question

My question is: take two random variables $X$ and $Y$. Does it hold that $$P(X<Y)=P(f(X)<f(Y))$$ For a monotonia funcion $f$ such as the exponential?

Thank you!

Answer 1

True or any strictly increasing function $f$, in particular for the exponential function.

Answer 2

Yes it's true, the reason is (for strictely increasing function $f$),$$X<Y\iff f(X)<f(Y),$$

and thus $$\{X<Y\}=\{f(X)<f(Y)\}.$$