I have been looking into equivalence relations and trying to figure out if certain relationships would be considered an equivalence relation. Lets using the following relation X between ordered pairs of natural numbers so that (a,b) is related to (c,d) but if and only if ad=bc;
[(a, b)X(c, d)]
Would this be considered an equivalence relation?
Lets check the definition:
Reflectivity: Let $(a,b) \in \Bbb{N} \times \Bbb{N} $. Since $ab=ba$ then $(a,b)X(a,b)$
Symmetry: Suppose $(a,b)X(c,d)$, this means $ad=bc$, since multiplication is commutative, $cb=da$ which gives by definition of $X$: $(c,d)X(a,b)$.
Transitivity: Suppose $(a,b)X(c,d)$ and $(c,d)X(e,f)$. This is $ad=bc$ and $cf=de$, which gives $f(ad)=f(bc)=b(cf)=b(de)$, therefore $d(fa)=d(eb)$ hence $fa=eb$ and therefore $(a,b)X(e,f)$.
Therefore X is an equivalence relation.
