"Reverse" quotients.

Question

Given a set $A$ and a equivalence relation $\sim$ on $A$, one can construct the quotient $A/{\sim}$. But what about the opposite? More precisely, given a set $X$, is there a way to know whether exists a set $A$ and a equivalence relation $\sim$ on $A$ such that $A/{\sim} = X$? If the answer is a definite yes, I suppose there must be a way to construct $A$ and $\sim$. Is there any material out there about this? I don't know the exact terms to look up, if there is any.

Answer 1

The following are all spiritually "the same thing":

1. Equivalence relations on $A$
2. Set partitions of $A$
3. Surjective functions $A\to X$

An equivalence relation on $A$ induces a set partition of $A$ into equivalence classes. This assignment has an inverse: given a set partition of $A$, define $\sim$ by $a\sim b$ iff $a,b$ are in the same part. Then, any set partition $\Gamma$ of $A$ induces a surjective function $A\to\Gamma$, taking an element $a$ to the part of $\Gamma$ containing $a$. This also is reversible: a surjection $A\to X$ induces a partition of $A$ into fibers, which are the preimages of singleton subsets of $X$.

Answer 2

If 2 elements of your set X have a common element, then X cannot be a quotient.

Else, let's write $X=\{Y, Y \in X \}$. Then Your A could be $\cup_{Y\in X} Y$, and your equivalence relation defined as $a \sim b$ iif "$a$ and $b$ both are elements of the same element of X".

Is it ok for you ?