I have a homework question that asks me to determine determine whether given relations are reflexive, symmetric, anti-symmetric, asymmetric or transitive.
One relation is:
$\rho \subseteq \mathbb \times \mathbb Z$, where $a~\rho~b$ if and only if there is $n \in \mathbb Z$ such that $a = bn$
The answer says it's reflexive and transitive. How come is it not anti-symmetric?
We only find $x~\rho~y$ and $y~\rho~x$ if $x = y$.
like in (1,1) or (2,2).
Take $a=0,b=5$. Then $a~\rho~b$ (take $n=0$) and $b~\rho~a$ (take $n=0$). However, $a\neq b$.
Antisymmetry requires that if $a~\rho~b$ and $b~\rho~a$ for some $a,b$, it must hold that $a=b$. Since the antecedent is true in the above example, but the consequent is not, $\rho$ cannot be antisymmetric.