Definition:Image (Set Theory)/Relation/Element/Singleton
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First Definition
Let $\mathcal R$ be a relation.
$\map {\mathcal R} s$ is the unique $t$ such that $s \mathcal R t$.
If $t$ is not unique, then $\map {\mathcal R} s = \O$.
That is, if $\map {\mathcal R} s = \set t$ where $\map {\mathcal R} s$ refers to the image of $s$ by $\mathcal R$ then:
- $\map {\mathcal R} s = t$ where $\map {\mathcal R} s$ now refers to the value of $\mathcal R$ at $s$.
Second Definition
To achieve this behavior, $z \in \map {\mathcal R} s$ if and only if $z \in t$ for the unique $t$ satisfying $s \mathcal R t$.
- $\map {\mathcal R} s = \set {z: \exists t: \paren {z \in t \land s \mathrel {\mathcal R} t} \land \exists! t: s \mathcal R t}$
Also denoted as
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 $\mathcal R$.
Historical Note
The first definition was first used by Bertrand Russell.
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 6.11$