# 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.

### Problem 2

Prove that the relation from Problem 1 is antisymmetric.

### 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.

### 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.