# Definition talk:Image (Set Theory)/Mapping/Element

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Why the union sign? The set defined is the element which is the second element of an ordered pair which is unique (by definition of mapping). What am I missing? --prime mover 18:02, 10 August 2012 (UTC)

- Without the union sign, it is the collection of
*all*valid second elements--namely, $\left\{ t \right\}$. It is the collection of all $t$ such that $( s,t ) \in \mathcal R$, so the resulting set has $t$ as its only member. I'm just using the union sign to make it functionally correct, even if the definition is not intuitive.

- To make the definition more intuitive, you could introduce (as per Whitehead/Russell or Quine) the notation $( \iota x P(x) )$ to mean "the sole $x$ such that $P(x)$". It could default to $\varnothing$ if $P(x)$ had no unique $x$ satisfying it. $f(s)$ would become $( \iota t ( s,t ) \in f )$. --Andrew Salmon 18:45, 10 August 2012 (UTC)