8.2.1 Elementary Examples of Relations

    Example 8.2.1.1.1 ▶ Square Roots

    Square roots are examples of relations:

    1. 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 $\sqrt{2}\not\in \mathbb {Q}$).

    • Example 8.2.1.1.2 ▶ Complex Logarithms

    The complex logarithm defines a relation

    \[ \log \colon \mathbb {C}\to \mathcal{P}\webleft (\mathbb {C}\webright ) \]

    from $\mathbb {C}$ to itself, where we have

    \[ \log \webleft (a+bi\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ \log \webleft (\sqrt{a^{2}+b^{2}}\webright )+i\arg \webleft (a+bi\webright )+\webleft (2\pi i\webright )k\ \middle |\ k\in \mathbb {Z}\right\} \]

    for each $a+bi\in \mathbb {C}$.

    Example 8.2.1.1.3 ▶ More Examples of Relations
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