Inverse Relations
The inverse, \(R^{1}\), of a binary relation, \(R:A\to B\), is the relation from \(B \to A\) defined by
In other words, you get the diagram for \(R^{1}\) from \(R\) by "reversing the arrows" in the diagram describing \(R\). Many of the relational properties of \(R^{1}\) correspond to different properties of \(R\). For example, \(R\) is total iff \(R^{1}\) is a surjection. How about the following relational properties?

\(R\) is a function iff \(R^{1}\) is

\(R\) is a surjection iff \(R^{1}\) is

\(R\) is an injection iff \(R^{1}\) is

\(R\) is a bijection iff \(R^{1}\) is