Definition:Image (Relation Theory)/Relation/Relation/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 $\displaystyle \prod_{i \mathop = 1}^n S_i$.

The image of $\mathcal R$ is the set defined as:
 * $\displaystyle \operatorname{Im} \left({\mathcal R}\right) := \left\{{s_n \in S_n: \exists \left({s_1, s_2, \ldots, s_{n-1}}\right) \in \prod_{i \mathop = 1}^{n-1} S_i: \left({s_1, s_2, \ldots, s_n}\right) \in \mathcal R}\right\}$

The concept is usually encountered when $\mathcal R$ is an endorelation on $S$:
 * $\displaystyle \operatorname{Im} \left({\mathcal R}\right) := \left\{{s_n \in S: \exists \left({s_1, s_2, \ldots, s_{n-1}}\right) \in S^{n-1}: \left({s_1, s_2, \ldots, s_n}\right) \in \mathcal R}\right\}$