Consider the set $A = \{1, 2, 3\}$, and set $B = A × A$. In the set $B$, consider then the relationship $C$ defined by placing
$(a, b) C (c, d)\iff a ≤ c$ and $b | d$
where $≤$ and $|$ denote respectively the usual arrangement and that of the division in $\mathbb{N}$.
• Prove that $C$ is an order relationship in $B$
Can someone explain me how to solve this exercise? Thanks.
$B$ consists of $9$ ordered pairs. To have a partial order you just need that if $e,f,g$ are ordered pairs in $B$, if $eCf, fCg$ then $eCg$ and that if $eCf, fCe$ then $e=f$. I suggest you find some pairs $eCf$ to get a feel for how this relation works. That may help proving that it is a partial order. It is not a total order because $(2,2)$ and $(3,3)$ are not comparable-neither is greater than the other.