Proofs with Relations and functions I need help with setting up a homework problem. I am having trouble finding where to start.
Problem: Suppose A is a set. Show that $i_A$ is the only relation on A that is both an equivalence relation on A and also a function from A to A.
First of all I'm not very clear what $i_A$ is. I think its a relation $_AR_A$? I'm really having trouble with the intuition behind proofs such as this. What proof strategy do I normally use to prove that ONLY ONE thing satisfies my conditions? Should I start my proof with $i_A$ or with the part about equivalence relation and function?
I'm also not very clear on what an equivalence relation is. I know that it has to be transitive, symmetric, and reflexive... but how does that help me?
I really want to learn all of this proof based math stuff but it is very counter intuitive to me. 

here are some hints.
First of all think about an equivalence relationship as a way to do a partition in a set (call it $A$). For a partition you can think about a sequence $\{A_i\}_i$ such that $A_i\cap A_j=\emptyset$ and $\cup A_i=A$.

Do you see how a partition is related with an equivalence relation? 
If you do: 
Now express a relation $a\equiv b$ as $(a,b)$. So $R=\{(a,b):a\equiv b\}$. Recall also that a function has that way but there is a restriction, i.e.  if $(a,b)\in R$ and $(a,c)\in R$ then $b=c$.

Now, you have all the equivalence relations as partitions. 

What happens if two elements of the set belong to the same $A_i$? 

What is the only partition that does not fit in the description of the question below?
