Say I have the function: $$x\,R\,y \iff y = 3^k$$ for some $k \in \mathbb Z$ and the set is:
$$A = \{1,1/3,1/27,1/4,3,1/36 , 2,2/9,9/4, 5\}$$
So in this scenario, how do I find the partitions of the set?
Knowing a fairly example should allow me to know how to do it in the future.
On
A partition of a set $A$ is a collection of $a_i \subset A$ such that each $a_i$ is disjoint from the others and $\bigcup a_i = A$.
An equivalence relation always partitions a set. Given a set $A$ and some relation $R$, the set of equivalence classes $A/R$ is a partition of $A$. Each member of $A/R$ is a subset of $A$ such that each member is $R$ related.
Since your given set is a finite set, perhaps the simplest approach would be to start with an element and determine all the other elements that are $R$ related to it. The set of all these elements is one equivalence class in the partition. Then pick another element from $A$ that is not in this first equivalence class, and form another class. Once all the elements of $A$ have been grouped, you will have formed the partition of $A$ that you are looking for.
An equivalence relation is always partition. In general you want to put it in the form of a set. Here, you can say,
$$A_{r}= \{(x,y) | y = \frac{x}{3^k}\}$$
Where $A_r$ denotes the equivalent partition corresponding to your relation. Also, $k$ is assumed to be an integer.