1.7 Binary 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

     a function a surjection an injection a bijection total none of the above an injection 

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

   Exercise 2

     a function a surjection an injection a bijection total none of the above total 

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

   Exercise 3

     a function a surjection an injection a bijection total none of the above a function 

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

   Exercise 4

     a function a surjection an injection a bijection total none of the above a bijection 
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