# 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. .

##### 1)

Hence, we have shown that and thus .

##### 2)

From , we have that .

Hence, . Thus we have shown .

From and it follows that .