Name for mappings where there is at least one y for every x There are names for several properties of mappings from $x$ in $X$ to $y$ in $Y$. I think we say that a mapping from X to Y is (a)...


*

*Function: there is at most one $y$ for every $x$

*Injective: there is at most one $x$ for every $y$

*Surjective: there is at least one $x$ for every $y$


How do we say "there is at least one $y$ for every $x$"?
 A: A relationship from X to Y where every element of X is mapped to an element of Y is called a "Total function".  Relationships where some elements of X are unmapped are called "Partial functions".
However formally all functions are total and people say "Total function" only where it would be ambiguous.
A: It is typical to define the domain of a function so that every element maps somewhere (a $y$ for every $x$), so there is no special name for it. For example, the function $f(x) = \frac{1}{x}$ is not properly a function from $\mathbb{R}$ to $\mathbb{R}$, but rather from $\mathbb{R} \setminus \{0\}$ to $\mathbb{R}$.
A: Mapping is the same thing as function.
A function $f$ from $X$ to $Y$ corresponds exactly one $y\in Y$ for every $x\in X$.
But there is also such a thing as set-valued function.
Notice that the sets $X$ and $Y$ are parts of the function $f$ (it's signature).
"At most one" in mathematics means that if there are two elements of something, then they have to be equal. While "at least one" means simply existence.
So, to answer the question, "there is at least one $y$ for every $x$" means there is a set-valued function, mapping each $x\in X$ into a non-empty set of $y\in Y$ (a subset of $Y$).
While your definition of a function actually describes a partial function (a more complex concept than just a function; as opposed to a "total" function).
