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 if and only if:

$\displaystyle \forall x := \tuple {x_1, x_2, \ldots, x_{n - 1} } \in \prod_{i \mathop = 1}^{n - 1} S_i: \forall y_1, y_2 \in S_n: \tuple {x, y_1} \in \mathcal R \land \tuple {x, y_2} \in \mathcal R \implies y_1 = y_2$

and

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

Thus, a mapping is an $n$-ary relation which is:

Many-to-one
Left-total, that is, defined for all elements in the domain.