Binary relations: The Lion quiz.

Question

This is my third question today and I think I'm abusing the platform a bit. In any case, here's the question:

Let $L$ be the total number of lions that live in Africa today. A binary relation $R$ is defined in $L$ as this: for every $p,q\in L$,$\;$ $pRq$ means that lion $p$ lives no more than 50km in distance from lion $q$. Prove that $R$ is a relation of equivalence (am I translating this correctly?) or isn't.

How does one go about something like this? It is apparent that it is, but what's the mathematical process of proving it?

Answer 1

The properties $R$ has to satisfy to be an equivalence relation, are:

• reflexivity: does any lion live mo more than 50km from himself ("near")?
• simmetricity: if a lion A lives near B, is it true that B lives near A?
• transitivity: if a lion B lives near A, and C lives near B, is it true that C lives near B?

You can go point by point, and when you find a condition that's not satisfied, then it's not an equivalence relation. If you can't find it, then it's an equivalence relation.

Answer 2

Take a piece of paper, and find three points $a,b,c$ such that the distance between $a$ and $b$ is at most one centimeter, and that the distance between $b$ and $c$ is also at most one centimeter. Is it always true that the distance between $a$ and $c$ is at most one centimeter? Look especially at cases were $a,b,c$ lie on a straight line...

If you can find points where the answer is "No", then the relation $R$ defined as "lies closer than one centimeter" cannot be an equivalence relation, because such a relation requires that if $a\, R\, b$ and $b\, R\, c$ then you always have $a\, R\, c$. Yet you just found a triple $a,b,c$ where the former are true but the latter is not...