What is this asking? Let = {(1,1), (2,1), (3,2), (4,3)}. Use the definition above to find , = 2, 3, 4, ….

Question

I'm having trouble as to seeing what this problem is asking. Am I supposed to create an algorithm that'll find $R^n$ for set R? If I'm just supposed to find $R^n$ for $R$, why does $n$ continue off into infinity?

Definition: Let $R$ be a relation on the set $A$. The powers $^$, $ = 1, 2, 3, \dots$ are defined recursively by $^1 = $ and $^{+1} = ^ \circ $. The definition shows that $^2 = \circ $, $^3 = R^2 \circ = ( \circ ) \circ $ and so on.

Let $ = \{(1,1), (2,1), (3,2), (4,3)\}$. Use the definition above to find $^$, $ = 2, 3, 4, \ldots$

Answer 1

Compute $R^2$ first. You have $1\to1,$ $2\to 1,$ $3\to 2$ and $4\to 3,$ so composing this with itself, $R^2 = \{(1,1),(2,1),(3,1),(4,2)\}.$ Now do it again, You see how this is going to go?

Answer 2

I think you are asking ``How am I supposed to solve an infinite number of problems?''

Without solving it for you, note that there are a finite number of relations over a finite set, so at some point $R^n$ must repeat itself with a fixed period (maybe even becoming a constant). Have you tried to solve it for some small values?