Let R be a relation on ZZ defined by (a,b),R,(c,d) if and only if ad-bc is divisible by 5.

Question

Let $R$ be a relation on $\mathbb{Z}	imes\mathbb{Z}$ defined by $(a,b)\,R\,(c,d)$ if and only if $ad-bc$ is divisible by 5. Then $R$ is:

Accepted Answer

Reflexive: $(a,b)R(a,b)$: $ab-ba=0$, divisible by 5. Holds.

Symmetric: if $ad-bc\equiv0\pmod5$, then for $(c,d)R(a,b)$ we need $cb-da\equiv0\pmod5$, which is just $-(ad-bc)\equiv0\pmod5$ --- true whenever $ad-bc\equiv0$. Holds.

Transitive: fails in general --- e.g. relating $(a,b)$ to $(c,d)$ to $(e,f)$ can lose the divisibility when the "denominators" $b,d,f$ are not all coprime to 5, so a valid counterexample can be constructed. Fails.

Solutions · 1