Definition:Image (Relation Theory)/Relation/Element/Singleton

First Definition
Let $\mathcal R$ be a relation.

$\mathcal R \left({s}\right)$ is the unique $t$ such that $s \mathcal R t$.

If $t$ is not unique, then $\mathcal R \left({s}\right) = \varnothing$.

That is, if $\mathcal R \left({s}\right) = \left\{ t \right\}$ where $\mathcal R \left({s}\right)$ refers to the image of $s$ by $\mathcal R$ then:


 * $\mathcal R \left({s}\right) = t$ where $\mathcal R \left({s}\right)$ now refers to the value of $\mathcal R$ at $s$.

Second Definition
To achieve this behavior, $z \in \mathcal R(s)$ if and only if $z \in t$ for the unique $t$ satisfying $s \mathcal R t$.


 * $\mathcal R \left({s}\right) = \left\{ z : \exists t: \left({ z \in t \land s \mathcal R t }\right) \land \exists ! t: s \mathcal R t \right\}$

Historical Note
The first definition was first used by Bertrand Russell.

Notation
Some authors distinguish the value of $\mathcal R$ at $s$ from the image of $s$ under $\mathcal R$ or image of set $A$ under $\mathcal R$ by denoting:


 * $\mathcal R ` s$ to be the value of $\mathcal R$ at $s$.


 * $\mathcal R " s$ to be the image of $s$ under $\mathcal R$.