Relations between (different) sets

Up to now we considered a relation \(R\) on a single set \(A\), viewed as a subset of the Cartesian Product \(R\subseteq A\times A\).

Sometimes we want to capture a relationship a different sort of relationship.

  • Consider the a relation between the integers \(\mathbb{Z}\) and the set \({0,1}\) where \(aR0\) if \(a\) is even and \(aR1\) if \(a\) is odd.

  • This can be expressed as a subset \(R\subseteq\mathbb{Z}\times\{0,1\}\). If we let \(E\) and \(O\) be the sets of even and odd numbers respectively, then \(R\) consists of the pairs \[(E\times\{0\})\cup (O\times\{1\}).\]

Another example.

\(S\) is the set of applicatnts for residency programs. \(R\) is the of residency programs.

We can construct a relation \(M\subseteq S\times R\) where \(sMr\) means that student \(s\) has a applied to program \(r\)

In this case the most natural picture might look like this.

One other example

Let \(S\) be a finite set and let \(\mathcal{P}(S)\) be its power set. We can define a relation \(R\) between \(S\) and \(\mathcal{P}(S)}\) by saying that \(sRx\) if \(s\in X\). What does this relation look like?