1.7 Binary Relations

Inverse Relations

The inverse, \(R^{-1}\), of a binary relation, \(R:A\to B\), is the relation from \(B \to A\) defined by

\(bR^{-1}a\; \text{ IFF }\; aRb\).

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?

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

    Exercise 1
  2. \(R\) is a surjection iff \(R^{-1}\) is

    Exercise 2
  3. \(R\) is an injection iff \(R^{-1}\) is

    Exercise 3
  4. \(R\) is a bijection iff \(R^{-1}\) is

    Exercise 4