The Clowder Project

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 $\mathcal{P}\webleft (B\webright )$, the relation $\mathord {\sim }_{\mathrm{triv}}$ is the function
   \[ \Delta _{\mathsf{true}}\colon A\to \mathcal{P}\webleft (B\webright ) \]

   defined by

   \[ \Delta _{\mathsf{true}}\webleft (a\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}B \]

   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 $b\in B$.