Lab 02 - Partial solution of Exercise 4
We will show that the first two statements are equivalent.
Let and .
Note that the full relation satisfies , hence the intersection is well-defined.
Then we claim that is the least relation satisfying . It is least, since by the properties of intersection. It remains to show that itself satisfies , i.e. that .
Now we claim , i.e. .
Hence, we have shown that and thus .
From , we have that .
Hence, . Thus we have shown .
From and it follows that .