Let's have a mapping relation $M$ between two sets. If $M$ is a function then we can check if it is injective and surjective. Is it valid to talk about these properties also if $M$ is not a function?
Example.
$A=\{1,2\}$, $B=\{a,b,c\}$, $M=\{(1,a),(1,b),(2,c)\}$.
Injectivity: $\forall y \in B: (x_1,y) \in M \land (x_2,y) \in M \Longrightarrow x_1 = x_2$
Surjectivity: $\forall y \in B \; \exists x \in A: (x,y) \in M$
So, could be $M$ called a bijective mapping relation? Does any other more appropriate term exist?
EDIT: In original question I used the term "mapping" but I wanted to describe relations.
$M$ is not a mapping at all. Rather, it's a relation satisfying some but not all of the defining properties of bijective mappings.
A relation between two sets $X$ and $Y$ is just any subset of $X\times Y$. There are a few important properties of such relations:
$R\subseteq X\times Y$ is functional iff $(x,y),(x,y')\in R\implies y=y'$.
$R$ is total iff for each $x\in X$ there is at least one $y\in Y$ such that $(x,y)\in R$.
$R$ is injective iff $(x,y),(x',y)\in R\implies x=x'$.
$R$ is surjective iff for each $y\in Y$ there is at least one $x\in X$ such that $(x,y)\in R$.
Note that functionality is the "dual property" to injectivity, and totality is similarly dual to surjectivity. The relation $M$ you describe is total, injective, and surjective, but not functional.