The trivial relation on $A$ and $B$ is the relation $\mathord {\sim }_{\mathrm{triv}}$ defined equivalently as follows:
-
1.
As a subset of $A\times B$, we have
\[ \mathord {\sim }_{\mathrm{triv}}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}A\times B. \] -
2.
As a function from $A\times B$ to $\{ \mathsf{true},\mathsf{false}\} $, the relation $\mathord {\sim }_{\mathrm{triv}}$ is the constant function
\[ \Delta _{\mathsf{true}}\colon A\times B\to \{ \mathsf{true},\mathsf{false}\} \]from $A\times B$ to $\{ \mathsf{true},\mathsf{false}\} $ taking the value $\mathsf{true}$.
-
3.
As a function from $A$ to $\mathcal{P}(B)$, the relation $\mathord {\sim }_{\mathrm{triv}}$ is the function
\[ \Delta _{\mathsf{true}}\colon A\to \mathcal{P}(B) \]defined by
\[ \Delta _{\mathsf{true}}(a)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}B \]for each $a\in A$.
-
4.
Lastly, it is the unique relation $R$ on $A$ and $B$ such that we have $a\sim _{R}b$ for each $a\in A$ and each $b\in B$.
-
1.
As a subset of $A\times B$, we have
\[ \mathord {\sim }_{\mathrm{cotriv}}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{Ø}. \] -
2.
As a function from $A\times B$ to $\{ \mathsf{true},\mathsf{false}\} $, the relation $\mathord {\sim }_{\mathrm{cotriv}}$ is the constant function
\[ \Delta _{\mathsf{false}}\colon A\times B\to \{ \mathsf{true},\mathsf{false}\} \]from $A\times B$ to $\{ \mathsf{true},\mathsf{false}\} $ taking the value $\mathsf{false}$.
-
3.
As a function from $A$ to $\mathcal{P}(B)$, the relation $\mathord {\sim }_{\mathrm{cotriv}}$ is the function
\[ \Delta _{\mathsf{false}}\colon A\to \mathcal{P}(B) \]defined by
\[ \Delta _{\mathsf{false}}(a)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{Ø} \]for each $a\in A$.
-
4.
Lastly, it is the unique relation $R$ on $A$ and $B$ such that we have $a\nsim _{R}b$ for each $a\in A$ and each $b\in B$.
-
1.
As a subset of $X\times X$, we have
\begin{align*} \mathord {\sim }_{\chi _{X}} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Delta _{X}\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ (x,x)\in X\times X\right\} .\end{align*} -
2.
As a function from $X\times X$ to $\{ \mathsf{true},\mathsf{false}\} $, we have
\[ \chi _{X}(x,y) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \mathsf{true}& \text{if $x=y$,}\\ \mathsf{false}& \text{if $x\neq y$} \end{cases} \]for each $x,y\in X$.
-
3.
As a function from $X$ to $\mathcal{P}(X)$, we have
\[ \chi _{X}(x)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ x\right\} \]for each $x\in X$.
-
1.
As a subset of $X\times X$, we have
\begin{align*} \mathord {\sim }_{\nabla _{X}} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\nabla _{X}\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}X\setminus \Delta _{X}\\ & = \left\{ (x,y)\in X\times X\ |\ x\neq y\right\} .\end{align*} -
2.
As a function from $X\times X$ to $\{ \mathsf{true},\mathsf{false}\} $, we have
\[ \nabla _{X}(x,y) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \mathsf{true}& \text{if $a\neq b$,}\\ \mathsf{false}& \text{if $a=b$} \end{cases} \]for each $x,y\in X$.
-
3.
As a function from $X$ to $\mathcal{P}(X)$, we have
\[ \nabla _{X}(x)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}X\setminus \left\{ x\right\} \]for each $x\in X$.
-
1.
Square Roots in $\mathbb {R}$. The assignment $x\mapsto \sqrt{x}$ defines a relation
\[ \sqrt{-}\colon \mathbb {R}\to \mathcal{P}(\mathbb {R}) \]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 $\sqrt{-}\colon \mathbb {Q}\to \mathcal{P}(\mathbb {Q})$ sends a rational number $x$ (e.g. $2$) to the empty set (since $\sqrt{2}\not\in \mathbb {Q}$).
The cotrivial relation on $A$ and $B$ is the relation $\mathord {\sim }_{\mathrm{cotriv}}$ defined equivalently as follows:
The characteristic relation $\chi _{X}$ on $X$ of Chapter 4: Constructions With Sets, Definition 4.5.3.1.1:
The antidiagonal relation on $X$ is the relation $\nabla _{X}$ defined equivalently as follows:
Partial functions may be viewed (or defined) as being exactly those relations which are functional; see Chapter 10: Conditions on Relations, Section 10.1.1.
Square roots are examples of relations:
The complex logarithm defines a relation
\[ \log \colon \mathbb {C}\to \mathcal{P}(\mathbb {C}) \]
from $\mathbb {C}$ to itself, where we have
\[ \log (a+bi)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ \log (\sqrt{a^{2}+b^{2}})+i\arg (a+bi)+(2\pi i)k\ \middle |\ k\in \mathbb {Z}\right\} \]
for each $a+bi\in \mathbb {C}$.
See [Wikipedia Contributors, Multivalued Function — Wikipedia, The Free Encyclopedia] for more examples of relations, such as antiderivation, inverse trigonometric functions, and inverse hyperbolic functions.
