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

Definition 1
Let $\RR$ be a relation.

$\map \RR s$ is the unique $t$ such that $s \mathrel \RR t$.

If $t$ is not unique, then $\map \RR s = \O$.

That is, if $\map \RR s = \set t$ where $\map \RR s$ refers to the image of $s$ by $\RR$ then:


 * $\map \RR s = t$ where $\map \RR s$ now refers to the value of $\RR$ at $s$.

Definition 2
To achieve this behavior, $z \in \map \RR s$ $z \in t$ for the unique $t$ satisfying $s \mathrel \RR t$.


 * $\map \RR s = \set {z: \exists t: \paren {z \in t \land s \mathrel \RR t} \land \exists! t: s \mathrel \RR t}$

Also denoted as
Some authors distinguish the value of $\RR$ at $s$ from the image of $s$ under $\RR$ or image of set $A$ under $\RR$ by denoting:


 * $\RR ` s$ to be the value of $\RR$ at $s$


 * $\RR {``} s$ to be the image of $s$ under $\RR$.

Historical Note
The first definition was first used by.