Definition:Image (Set Theory)/Relation/Element/Singleton
... if not rewritten completely
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First Definition
Let $\mathcal R$ be a relation.
The below definition is only applicable when/if $\mathcal R$ is many-to-one. The statement therefore needs to be qualified. The second definition below captures this fact, but this page needs to be rethought. It is also worth considering whether this material is adequately covered as a special case of Definition:Direct Image Mapping of Mapping.
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$\map {\mathcal R} s$ 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.
Where?
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Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 6.11$
