Definition:Mapping/General Definition

Definition
Let $\displaystyle \prod_{i \mathop = 1}^n S_i$ be the cartesian product of sets $S_1$ to $S_n$.

Let $\displaystyle \mathcal R \subseteq \prod_{i \mathop = 1}^n S_i$ be an $n$-ary relation on $\displaystyle \prod_{i \mathop = 1}^n S_i$.

Then $\mathcal R$ is a mapping iff:
 * $\displaystyle \forall x := \left({x_1, x_2, \ldots, x_{n-1}}\right) \in \prod_{i \mathop = 1}^{n-1} S_i: \forall y_1, y_2 \in S_n: \left({x, y_1}\right) \in \mathcal R \land \left({x, y_2}\right) \in \mathcal R \implies y_1 = y_2$

and
 * $\displaystyle \forall x := \left({x_1, x_2, \ldots, x_{n-1}}\right) \in \prod_{i \mathop = 1}^{n-1} S_i: \exists y \in S_n: \left({x, y}\right) \in \mathcal R$

Thus, a mapping is an $n$-ary relation which is:
 * Many-to-one
 * Left-total, that is, defined for all elements in the domain.