The Clowder Project

A relation $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ from $A$ to $B$^1,2 is equivalently:

1. A subset $R$ of $A\times B$.
2. A function from $A\times B$ to $\{ \mathsf{true},\mathsf{false}\} $.
3. A function from $A$ to $\mathcal{P}\webleft (B\webright )$.
4. A function from $B$ to $\mathcal{P}\webleft (A\webright )$.
5. A cocontinuous morphism of posets from $\webleft (\mathcal{P}\webleft (A\webright ),\subset \webright )$ to $\webleft (\mathcal{P}\webleft (B\webright ),\subset \webright )$.