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Relations Between Sets 193
Diagrams for relations from A to B differ from diagrams for relations
on A. Since there are two sets A and B in a relation from A to B, we have
to draw labeled nodes for each of the two sets. Then we draw arrows from x
to y whenever xR y. The following figure illustrates this for Example 11.16.
A
1
2
B
{
1
}
{
2
}
{
1,2
}
Figure 11.3. A relation from A to B
The ideas from this chapter show that any relation (whether it is a
familiar one like ≥, ≤, =, |, ∈ or ⊆, or a more exotic one) is really just a set.
Therefore the theory of relations is a part of set theory. In the next chapter,
we will see that this idea touches on another important mathematical
construction, namely functions. We will define a function to be a special
kind of relation from one set to another, and in this context we will see
that any function is really just a set.