1.3 Relations

Let us turn our attention to a more difficult problem. Consider graphing a relation, such as

The procedure taught is to first graph the boundary; the boundary of is given by s(x,y) = t(x,y). Our example's boundary is given by

We then shade in the appropriate side.

It may seem that graphing relations is not much harder than graphing equations, given the simple approach outlined earlier. But consider the two relations

Both have the same boundary, which does not break the plane nicely into two ``sides''.

The two relations have different graphs, one of which may be given by our side-testing procedure.

If this is not troubling enough, consider the relation

which, again, has the same boundary as the two earlier relations xy=1 and . The graph of this new relation follows:

The graph contains all points that satisfy

The only true relationship between the boundary of a graph and the actual graph is that of containment: the graph contains its boundary. In the case of strict inequality, the graph does not contain its boundary.

It appears that graphing relations is indeed more difficult than graphing equations. However, this is not the case; consider the following relation:

This relation may be expressed as an equation, as follows:

We have not chosen a simpler problem: we have, however, illustrated some of its hidden difficulties. These difficulties are nicely illustrated in the next example.

A single graph may contain zero, one, and two-dimensional elements; consider the equation

whose graph follows:

The graph contains all point which satisfy

It seems that the entire idea of generating graphs as collections of lines is fundamentally flawed, as a graph may contain two-dimensional elements. Representing two-dimensional elements with a collection of lines is inefficient at best, and simply unconscionable at worst.

Jeff Tupper

March 1996