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.


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.


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.