Square roots are examples of relations:
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1.
Square Roots in $\mathbb {R}$. The assignment $x\mapsto \sqrt{x}$ defines a relation
\[ \sqrt{-}\colon \mathbb {R}\to \mathcal{P}\webleft (\mathbb {R}\webright ) \]from $\mathbb {R}$ to itself, being explicitly given by
\[ \sqrt{x}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} 0 & \text{if $x=0$,}\\ \left\{ -\sqrt{\left\lvert x\right\rvert },\sqrt{\left\lvert x\right\rvert }\right\} & \text{if $x\neq 0$.} \end{cases} \] -
2.
Square Roots in $\mathbb {Q}$. Square roots in $\mathbb {Q}$ are similar to square roots in $\mathbb {R}$, though now additionally it may also occur that $\sqrt{-}\colon \mathbb {Q}\to \mathcal{P}\webleft (\mathbb {Q}\webright )$ sends a rational number $x$ (e.g. $2$) to the empty set (since $\sqrt{2}\not\in \mathbb {Q}$).
