How to mathematically show that the relation is transitive?

Question

Problem:
Show that the relation $x R y$ iff $x \leq y$ is a poset over the set of integers $\mathbb{Z}$

My work:
I know that to show the relation is a poset or a post order, I have to show the relation is reflexive, transitive, and anti-symmetric.

For reflexive, suppose $a$ is an integer in $\mathbb{Z}$, then $a$ is by definition $\leq a$. Therefore $(a,a)$ is an ordered pair in the relation or $a R a$.
For anti-symmetric, suppose $a$ and $b$ are some integers in $\mathbb{Z}$. If $a R b$, $a \leq b$. The only way $b R a$ or $b \leq a$ is if $b = a$, in that case $b \leq a$. If $b>a$, then $a$ cannot be $\geq b$. We proved that if $a R b$ and
$b R a$, then $a =b$.

I had trouble with transitive. I knew this was transitive right away because lets say that
$1 \leq 2$, $2 \leq 3$, then you can definitely conclude that $1 \leq 3$. How would you mathematically explain this? If $a \leq b$, $b \leq c$, then $a \leq c$. Is there some mathematical way to say $a \leq c$? I would just say by intuition.

Answer 1

How would you mathematically explain this? If $a≤b$, $b≤c$, then $a≤c$. Is there some mathematical way to say $a≤c$?

That's all you need to say, really.

For any integer triplet $(a, b, c)$, if $a\leq b$ and $b\leq c$, then $a\leq c$.   Thus $(\Bbb Z, \leq)$ is transitive.

$$\forall (a,b,c)\in\Bbb Z^3\;\Big((a\leq b)\wedge (b\leq c) \;\to\; (a\leq c)\Big)$$

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If you like, you can demonstrate this by considering the four cases:

• If $a<b$ and $b<c$ then $a<c$.
• If $a<b$ and $b=c$ then $a<c$.
• If $a=b$ and $b<c$ then $a<c$.
• If $a=b$ and $b=c$ then $a=c$.

Answer 2

Assume that $aRb$ and $bRc$, this means that $a \leq b$ and $b \leq c$. Let's examine this by cases. Either $a < b$ or $a = b$.

If $a < b$, then if $b = c$, we have immediately that $a < c$. If $b < c$, then $a$ must also be smaller than $c$, so $a < c$.

If $a = b$, then if $b = c$, we have $a = b = c$. If $b < c$, then it again follows immediately that $a < c$.

Thus we can conclude that $a \leq c$.