Equivalence classes "mod" a function Suppose I have a function, and I want to talk about the sets of points in the domain of the function that map to the same point in the image under the function. In general, what's a good term for this? Specifically, I have a hash function, and I want to talk about the equivalence classes "mod" the hash function, in a CS context. I know quotient spaces are relevant here, but that is specific to abstract algebra, and I want a term or expression that would be universally understood.
 A: What you describe is just the ordinary inverse image $f^{-1}$. For instance, if $f(x) = x^2$ and $x$ is real then $f^{-1}(y) = \{\pm \sqrt{y}\}$. You don't need equivalence classes or quotients. 
In general, if $A$ and $B$ are sets and $f: A \rightarrow B$ is a function between them, there is always a function $f: P(A) \rightarrow P(B)$ defined by $f(U) = \{f(x) \ | \ x \in U\}$ for any $U \in P(A)$. Similarly there is an inverse image $f^{-1}: P(B) \rightarrow P(A)$ defined by $f^{-1}(V) = \{x \in A \ | \ f(x) \in V \}$ for any $V \in P(B)$. 
A: For a function $f : X \to Y$ the terminology is


*

*for a subset $B$ of $Y$, $f^{-1}(B) = \{x \in X : f(x) \in B\}$ is the preimage or inverse image of $\pmb B$

*for a singleton set $\{y\}$ the preimage $f^{-1}(\{y\})$ is also denoted $f^{-1}(y)$ and is also called the fibre above $\pmb y$ or the level set of $\pmb y$. In American English "fiber" is the more common spelling.
I believe "fibre" is more common for discrete objects such as groups and vector spaces and "level set" for real-valued functions.
A: I agree with the other answers (preimage or inverse image), but in the context of hash functions in computer science you could use the term collision.
For instance, quoting this other Wikipedia entry:

Collision resistance is a property of cryptographic hash functions: a
  hash function is collision resistant if it is hard to find two
  inputs that hash to the same output;

