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

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Definition 1

Let $\RR$ be a many-to-one 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, $z \in \map \RR s$ if and only if $z \in t$ 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 Bertrand Russell.

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Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 6.11$