Transitive closures

May 2010

I'm trying to find this example since 4 hours, but can't find one. I'm starting thinking it's impossible...

Find an example of relations \(\displaystyle R_1\) and \(\displaystyle R_2\) on some set \(\displaystyle A\) such that, if we let \(\displaystyle R = R_1 \setminus R_2\) and we let \(\displaystyle S_1\), \(\displaystyle S_2\) and \(\displaystyle S\) be the transitive closures of \(\displaystyle R_1\), \(\displaystyle R_2\) and \(\displaystyle R\) respectively, then \(\displaystyle S_1 \setminus S_2 \not\subseteq S\) and \(\displaystyle S \not\subseteq S_1 \setminus S_2\).