Definition of "quotient set"

Question

I searched and searched about quotient set and cannot figure out what it is. At the beginning, I thought it was the same as partitions, but now I'm confused. Can someone show some examples and explain?

Answer 1

A quotient set is what you get when you "divide" a set $A$ by $B\subseteq A$, wherein you set all elements of $B$ to the identity in $A$. For example, if $A=\Bbb Z$ and $B=\{5n\mid n\in\Bbb Z\}$, then you're making all multiples of $5$ zero for all intents and purposes, so the quotient is $\{0,1,2,3,4\}$.

Another (and more correct) way of saying this is that a quotient set is all equivalence classes on the set $A$ under a given equivalence relation. In the example above, $aRb\iff 5|(a-b)$, so clearly the equivalence classes are $n\equiv 0,1,2,3,4\pmod 5$. In reality, you can select any number from each equivalence class, so $\{20,-34,77,63,-1\}$ would be a "correct" quotient set, just not canonical.