Let $R$ be an equivalence relation on $A$...

Question

knowing that in $A$ is more than three elements. we define $R^c=A\times A \setminus R$, then which of these statements is surely correct?
a) if $R^c$ is transitive, then $R$ has only one equivalence class.
b) if $R^c$ is symmetric, then $R$ has only one equivalence class.
c) if $R^c$ is irreflexive, then $R$ has only one equivalence class.
d) if $R^c$ is transitive, then $R$ is the identity relation.
e) if $R^c$ is symmetric, then $R$ is the universal relation (full relation $R=A\times A$).

I am really trying to find the purpose or what should I understand to solve this question, I might be able to track this question while writing an example set $A$ and writing a relation with $16$ elements and start trying to disprove each one of the options, but that really made me just complicate myself a little and saw that there was a high chance I'll make lots of mistakes missing elements etc.. I was wondering, there must be some better way to solve this question, since if the question asked for $10$ elements, my approach won't work and I will hesitate to check with less elements than $10$.
I can't understand the idea of one equivalence class, I understand the definition of equivalence class, but seems like not deep enough.

I would really appreciate to see approaches from more experienced people in this question and how would you solve it, I appreciate all the help, thanks in advance.
Final answer is (a).