Question

Let ‘R’ be a relation defined on a set of integers Z as follows:

∀ 𝑎, 𝑏 ∈ 𝑍, 𝑎𝑅𝑏 iff 𝑏 = a^r

for some integer 𝑟. Show that R is a partially

ordered relation.

Answer 1

$R$ is a partially ordered relation if and only if it is reflexive, antisymmetric and transitive.

1) reflexive

$aRa,$ because $a=a^1.$

2) antisymmetric

If $aRb$ and $bRa,$ then $b=a^m$ and $a=b^n$ for some integers $m$ and $n.$ 

So, $b=a^m=(b^n)^m=b^{nm}$ and it means that $mn=1.$

There are two cases: $m=n=1$ and $m=n=-1.$

If $m=n=1,$ then $a=b.$

If $m=n=-1,$ then it is possible only for $a=b=\pm1.$

3) transitive

If $aRb$ and $bRc,$ then $b=a^m$ and $c=b^n$ for some integers $m$ and $n.$

Since $c=b^n=(a^m)^n=a^{mn},$ we have that $aRc.$