In a discrete mathematics course, I stumbled upon the following problem. I have an idea how to solve the problem based on the fact that the power of a relation repeats after 3 consecutive powers; that is $R^1=R^4, R^2=R^5, R^3=R^6$, and so on. However, I cannot put it into words on a formal way. How would you approach it?
Problem
Let $R$ be the relation on the set of people with doctorates such that $(a, b) \in R$ if and only if $a$ was the thesis advisor of $b$. (You may assume that every person with a doctorate has a thesis advisor.)
- When is an ordered pair $(a,b)$ in $R^2$?
- When is an ordered pair $(a,b)$ in $R^n$, when $n$ is a positive integer?
The problem is asking for an intepretation of when $(a,b)\in R^2$, and when $(a,b)\in R^n$, i.e. it is asking for an explanation in terms of people with doctorates and advisors.
That $(a,b)\in R^2$ means that there exists some $c$, such that $(a,c)\in R$ and $(c,b)\in R$. This means that $a$ was the thesis advisor of someone who was the thesis advisor of $b$.
Similarly, $(a,b)\in R^n$ means that $a$ was the thesis advisor of someone, who was the thesis advisor of someone who ... ($n-1$ times) ... who was the thesis advisor of $b$. You can say that $a$ is the mathematical great grandfather of $b$ in $n$ generations.