Inverse Relations and Inverse Functions
The identity function
Definition: Let \(A\) be a set. The identity function \(i_A:A\to A\) is the function defined by \(i_A(x)=x\) for all \(x\in A\).
As a set of ordered pairs, \(i_A\subset A\times A\) consists of all pairs \((a,a)\) for \(a\in A\).
The identity function
Proposition: The identity function is bijective.
The inverse of a relation
Definition: Let \(A\) and \(B\) be sets and let \(R\) be a relation on \(R\subset A\times B\). The inverse relation \(R^{-1}\) to \(R\) is the relation on \(B\times A\) defined by \[ R^{-1} = \{(b,a)\in B\times A: (a,b)\in R\}. \]
Examples of inverse relations
Example: Let \(A=\mathbb{R}\) and let \(R\) be the relation \(<\). Then \(R\) consists of all pairs \((a,b)\in\mathbb{R}\times\mathbb{R}\) with \(a<b\). The inverse relation \(R^{-1}\) consists of all pairs \((b,a)\in\mathbb{R}\times\mathbb{R}\) with \((a,b)\in R\). Thus \(R^{-1}\) is the relation \(>\).
Another example
Example: Let \(A=\mathbb{Z}\) and let \(R\) be the relation “divides”, so that \(R\) consists of pairs \((a,b)\in\mathbb{Z}\times\mathbb{Z}\) where \(a|b\). The inverse relation \(R^{-1}\) consists of pairs \((a,b)\) where \(b|a\), or, in other words, where \(a\) is a multiple of \(b\).
So the inverse relation to \(a|b\), meaning \(a\) is a divisor of \(b\), is the relation \(aRb\) when \(a\) is a multiple of \(b\).