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