Discuss the Transitivity of Binary Relations
$\mathcal{S} $ $a$ on $\Bbb R $ defined by $a (x, y)$ $\in \Bbb R^2 $--> $x \leq ay$ ( for some a $ \in \Bbb R$ )
I have this assignment about transitivity and binary relation, but i have no idea how can it be related by that formula on top.
I am not quite sure about this notation but suppose that $R$ is the relation such that if $(x,y)\in R$ then $x\leq ay: a,x,y \in \mathbb{R}$.
Let's think about $y\geq \dfrac{x}{a}$. Geometrically, this is interpreted as the region including and above the line $y=x/a$.
So suppose $(x,y) \in R$ and $(y,z)\in R$. We want to show that $(x,z)\in R$, hence the transitivity of the relation.
If $(x,y)\in R$, then $y\geq\dfrac{x}{a}:a\in \mathbb{R}$ and if $(y,z)\in R$, then $z\geq\dfrac{y}{b}:b\in\mathbb{R}.\quad$
Now, we want to show that $z\geq\dfrac{x}{c}:c\in\mathbb{R}$.
Well, since $z\geq\dfrac{y}{b}$ and $y\geq\dfrac{x}{a}$, then $z\geq\dfrac{x}{ab}$
Since $a,b\in\mathbb{R}$, $ab\in\mathbb{R}$ and we can replace the product with $c\in\mathbb{R}$.
Thus, the relation is transitive.