Let a and b be two integers. The relation $\alpha$ $\subseteq$ ZxZ is defined by x $\alpha$ y iff аx+by=1. Find all possible integers a and b for which $\alpha$ is nonempty.
For all а and b proof existence or absence of each of following properties: irreflexive, symmetric, antysymmetric and transitive.
Can somebody help with this problem?
There exist $x,y \in \mathbb{Z}$ such that $xa+yb=1$ if and only if $\gcd(a,b)=1$. The implication from left to right is an easy exercise, the one from right to left follows from euclid's algorithm. If you do not know this, it basically provides a way to compute $\gcd(a,b)$ by subsequently applying division with remainder. So the pairs $(a,b)$ for which the relation is nonempty are exactly the ones for which $\gcd(a,b)=1$.