# Relations study problems

### Problem 1

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.

hints

annotated solutions

### Problem 2

Prove that the relation from Problem 1 is antisymmetric.

annotated solutions

### Problem 3

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.

hints

solution to (a)

solution to (b)

### Problem 4

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.

hints

solution