Let $R$ be a relation on Cartesian product $\mathbb N \times \mathbb N$ where $(x,y)\mathrel R(u,v)$ iff $xv - yu = 1$.

Question

I am stuck on the transitive proof, as I'm kinda unable to conclude from there on.

This is what I have done so far:

-Reflexive : $(x,y)\mathrel R(x,y) \implies xy - yx = 1$

Hence $R$ is not reflexive.

-Symmetric: $(x,y)\mathrel R(u,v) \implies xv - yu = 1$

$\implies-(uy - vx) = 1$

$\implies uy - vx = -1$

Hence R is not symmetric.

-Transitive:

$(x,y)\mathrel R(u,v) \implies xv - yu = 1 \tag{$1$}$

$(u,v)\mathrel R(w,p) \implies up - vw = 1 \tag{$2$}$

I am stuck here. I'm hesitant to just conclude that it's not transitive, so how could I go on about proving that it's transitive from here on? Thanks

Answer 1

$(3,2)R(1,1)$ because $3-2=1$

$(1,1)R(2,3)$ because $3-2=1$

But $(3,2)$ is not related to $(2,3)$ since, $(3\times 3)-(2\times 2)=5≠1$