Good afternoon,
I am having trouble defining the following:
Let $R$ denote the binary relation $\ge$ on the set of Natural Numbers $\mathbb N$. Find $R^2$.
What I have so far for R:
$R= \{\langle a,b\rangle\mid a \in R \land b \in R \land (a\ge b) \}$
Inductive clause: For numbers $x$ and $y$ in $R$, if $\langle x,y\rangle$ is in $R$, then $\langle x+1,y\rangle$ is in $R$ and $\langle x+1, y+1\rangle$ is in $R$. Extremal clause: Nothing is in R unless it is obtained from the Basis and Inductive clause.
Thanks, David
HINT: Always start with the definition(s); at this level that’s often all that you need. Let $m,n\in\Bbb N$; then by definition $\langle m,n\rangle\in R^2$ if and only if there is a $k\in\Bbb N$ such that $\langle m,k\rangle\in R$ and $\langle k,n\rangle\in R$. Now rewrite this using the more familiar $\le$ instead of $R$, and see what it tells you about $m$ and $n$.