Inverse Relations
For the discussion of relations, it is helpful to review two standard definitions. These definitions can be found, for example, in [2].
Definition: Relation
In the definition above, [latex]S[/latex] and [latex]T[/latex] may be any sets.
Definition: Inverse of a Relation
If [latex]R[/latex] is a relation from the set [latex]S[/latex] to the set [latex]T[/latex], then the inverse of [latex]R[/latex] is the relation from the set [latex]T[/latex] to the set [latex]S[/latex] given by:
\begin{equation}
R^{-1} = \{(t, s) \: | \: (s, t) \: \epsilon \: R \}.
\end{equation}
Accordingly, one passes to the inverse of a relation by interchanging the two values in each pair. Let us see what this implies for relations as discussed in a basic algebra course.
Suppose that [latex]S[/latex] and [latex]T[/latex] are sets of real numbers and consider the example
\begin{equation}
R = \{(x,y)\ |\ 2x+3y = 6\}.
\end{equation}
Using the standard definition above,
\begin{equation}
R^{-1} = \{(y,x)\ |\ (x,y)\ \epsilon\ R\} = \{(y,x)\ |\ 2x+3y= 6\}.
\end{equation}
The definition of the inverse does not call for an interchange of the variable names; it requires the interchange of the two values in each pair. Interchanging the two values in each pair is not the same thing as interchanging the two variable names.
For the graph of the inverse in this example, the original [latex]y[/latex]-values are now in the first coordinate position, so [latex]y[/latex] has become the domain variable. Therefore, the horizontal axis should be labeled [latex]y[/latex], and the [latex]y[/latex]-values should be plotted with reference to that axis. The original [latex]x[/latex]-values are now in the second coordinate position, and [latex]x[/latex] has
become the range variable. Therefore, the vertical axis should be labeled [latex]x[/latex], and the
[latex]x[/latex]-values should be plotted with reference to that axis.
When the function is considered as a special case of a relation, the treatments
described above for functions and relations are consistent.
