Define a relation \(\ll\) on \(\mathbb{Z}^2\) by \((a,b) \ll (c,d)\) if and only if either \(a < c\) or else \(a=c\) and \(b \leq d\). Prove that \(\ll\) is transitive.
Prove that the relation from Problem 1 is antisymmetric.
A circle is determined by its center (x,y) and its radius r. So we can represent a circle as a triple of real numbers (x,y,r), where r is positive. (We won't include circles of radius zero!) Let C be the set containing such triples, i.e. \(C = \mathbb{R} \times \mathbb{R} \times \mathbb{R}^+\).
Let's define the relation R on C by
(x,y,r) R (a,b,s) if and only if \(\sqrt{(x-a)^2 + (y-b)^2} \le s- r\)
(a) Explain in words what it means for the circle represented by (x,y,r) to be related to the circle represented by (a,b,s).
(b) Prove that this relation is antisymmetric.
Define a relation \(\thicksim\) on the set of all functions from \(\mathbb{R}\) to \(\mathbb{R}\) by the rule
\(f \thicksim g\) if and only if there is a \(k \in \mathbb{R}\) such that f(x) = g(x) for every \(x \geq k\).
Prove that \(\thicksim\) is an equivalence relation.