Let us say we have two probability measures, $X$ and $Y$ on sample spaces $\Sigma\_X$ and $\Sigma\_Y$ (which are finite sets, with the largest sigma algebra on each space) and suppose we get measure $Y$ via a function $f: g(X) \rightarrow Y$. What is the [Relative Entropy](https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence) $D\_{KL}(X || Y)$ when g is:


1. An isomorphism
2. A Surjection, non-isomorphism
3. An injective non-surgective



What is the Relative Entropy between distributions $X$ and $Y$, when $Y$ is a function of $X$?
As R W noted, the relative entropy depends only on the individual (marginal) distributions of $X$ and of $Y$; it does not depend on the joint distribution of $X$ and $Y$. So, the condition that $Y$ is a function of $X$ is quite irrelevant as far as the relative entropy is concerned, not matter what the function is. 


Perhaps you mean the conditional (rather than relative) entropy 
$$H(Y|X):=-E\ln p(Y|X)$$
of $Y$ given $X$, where $p$ is the conditional probability mass function (pmf) of a random variable $Y$ given $X$ (assuming that the conditional distribution of $Y$ given $X$ is discrete). 


If so, then (as you guessed in the original version of your post) [$H(Y|X)=0$ if $Y$ is a function of $X$](https://en.wikipedia.org/wiki/Conditional_entropy#Conditional_entropy_equals_zero) , because then the conditional pmf $p$ is supported on a singleton set, for each given value of $Y$. 


