The Clowder Project

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

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

   defined by

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

   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 $b\in B$.