For me, quotient set/group seems to be a impossible math notion which can be understood. So, I would like to ask you if you have some books, very suggestive examples, links where I can find out more about this kind of sets. I have been trying to learn this topic for two years, but I did not managed to understand the true meaning of the word quotient set.
Thanks!
Let me suggest one approach that sometimes works when trying to learn difficult abstract topics. Often (not always and maybe not often enough) a difficult topic is introduced using lots of motivational examples. This can give a gentler approach that focuses on learning the intuitiveness behind the topic. But for some this just obscures what is going on.
So, maybe the following could help you. Try to just learning first the definition of a coset. Learn then the definition of a normal subgroup. Then learn the definition of a quotient group. Focus just on memorizing the definition paying only attention to the formalism of the definitions. Get to a point where you can neatly and perfectly write down these definitions.
After this find an example and learn this one example. Start with $A_3\leq S_3$. Find all the cosets. Just write them down and don't worry about what they "really are" or what the deeper meaning is. The quotient is then the set/group of all these cosets. Try to make a multiplication table (Cayley table) for the group. Again, just treat it as a formal game).
Next, consider the normal subgroup $n\mathbb{Z}$ of the group $\mathbb{Z}$ (both additive/Abelian). The try to prove that the order of the quotient is $n$.
I know some will argue against this approach to learning a topic, but if you have spent two years, then this might work (unless this is what you have already been trying). Just an idea that has worked for me in the past.