Relations Problem 1

First, notice that your base set is \(\mathbb{Z}^2\). That is, each object in the base set is a 2D point with integer coordinates. The relation \(\ll\) relates a 2D point to another 2D point.

Now, recall what it means for a relation R on a set A to be transitive:

For any elements s,t,w in A, if sRt and tRw, then sRw.

For this problem, the elements s, t, w are all 2D points. So your proof will be manipulating three 2D points. E.g. p in the above definition would become something like (a,b) in your proof.

Now write out the hypothesis sRt and tRw using 2D points and the relation name \(\ll\). Same with the conclusion sRw.

Finally, you'll need to apply the definition of \(\ll\) to the information in both the hypothesis and the conclusion. You should now be left with an algebra problem to solve.