# 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