Let X={ (m,n) ∈ Z×Z, n \neq 0 } Let R be the relation on X defined by ((m,n),(m’,n’)) ∈ R iff mn’=m’n. (a) Prove that R is an equivalence relation on X. (b) What happens if we allow n=0 in X?
I got the first question fine but I'm not able to understand the repercussions if n=0 were to be allowed. I mean m can have values of 0 anyways so how would is possibly affect the resonating?
Suppose you allow $n=0$. Then $((1,0), (0,0)) \in R$ and $((0,0), (0,1) \in R$. By transitivity, $((1,0), (0,1)) \in R$ implying $1 \cdot 1 = 0 \cdot 0$, a contradiction. This means that we lose transitivity by allowing $n=0$ and thus $R$ is no longer an equivalence relation.