The Clowder Project

The trivial relation on $A$ and $B$ is the relation $\mathord {\sim }_{\mathrm{triv}}$ defined equivalently as follows:

1. 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. 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. 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. 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 $b\in B$.

The cotrivial relation on $A$ and $B$ is the relation $\mathord {\sim }_{\mathrm{cotriv}}$ defined equivalently as follows:

1. 1.

   As a subset of $A\times B$, we have

   \[ \mathord {\sim }_{\mathrm{cotriv}}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{Ø}. \]
2. 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. 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 $a\in A$.

The characteristic relation $\chi _{X}$ on $X$ of Chapter 4: Constructions With Sets, Definition 4.5.3.1.1:

1. 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. 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. 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 $x\in X$.

The antidiagonal relation on $X$ is the relation $\nabla _{X}$ defined equivalently as follows:

1. 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. 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. 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 $x\in X$.

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:

1. 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. 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 complex logarithm defines a relation

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

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.