Partial order relation (Antisymmetric property), given a relation $xRy \iff x-y\le 4$

Question

Given the set: $A=\{1,2,3,\dots,19,20\}$. The relation $R$ is defined on $A$ as: $xRy\Leftrightarrow x-y\leq4$

Is $R$ a partial order relation?

I know that for a relation to be partial order it has to be reflexive, antisymmetric and transitive.

I do not know how to show that $R$ is antisymmetric. For starters I do know this: $$\left.\begin{matrix} aRb\\ bRa \end{matrix}\right\} \Rightarrow a=b $$ But I do not know how to apply this to the given relation $R$.

What I've tried is this: $$\left.\begin{matrix} aRb\\ bRa \end{matrix}\right\}\Rightarrow \left.\begin{matrix} a-b\leq4\\ b-a\leq4 \end{matrix}\right\}\Rightarrow \left.\begin{matrix} a\leq4+b\\ b\leq4+a \end{matrix}\right\}$$ But I'm stuck at expanding this further, is this the correct method?

Thanks in advance.

Answer 1

I think this is not a partial order. Indeed the antisymmetry does not work. Consider

$$a=2, \quad b=4$$

Then you have $aRb,$ since $2-4 = -2 \le 4$, and $bRa$, since $4-2 =2\le 4$.