Order relation proof ...

Question

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 $B$

I did this:

1. $C$ is reflexive. For all $(a,b)\in\mathbb{R}\times\mathbb{C}$ it holds $(a,b)R(a,b)$ because $a\leq a$ and $b\leq b$;
2. If $(a,b)R(x,y)$ and $(x,y)R(a,b)$, then both $a\leq x,b\leq y$ and $x\leq a,y\leq b$. It follows $a=x$ and $b=y$, so $(a,b)=(x,y)$
3. Let $(a,b)C(x,y)$ and $(x,y)C(c,d)$. Hence, $a\leq x$ and $x\leq c$, so $a\leq c$. The same for $b\leq d$. Both imply $(a,b)C(c,d)$.

Answer 1

You need to show that $(a, b) C (c, d)\iff a ≤ c$ and b|d.

1. $a\leq a$ is a correct argument. But you need to show that $b | b$ (which is true, because $b=1\cdot b$).
2. Same here: $a=x$ is correct. Now try to prove that if $b|d$ and $d|b$, then $b=d$.
3. Same: $a\leq c$ is correct. Now try to prove that if $b|y$ and $y|d$, then $b|d$.

Use the definition that $a|b$ if there exists $n\in\mathbb{Z}$ such that $an=b$.