Is the relation $x \geq 2y$ transitive?

Question

I am trying to understand if the relation $x \geq 2y$ is transitive. I think the answer is no for the following reasons. Can someone please let me know if I am correct or incorrect. If incorrect, why am I incorrect.

Definition of transitive: If $\left ( x,y \right )$ are elements of the relation, and $\left ( y,z \right )$ are elements of the relation, then this implies that $\left ( x,z \right )$ are elements of the relation.

Here is a counter example: $\left ( 4,2 \right )$ are elements of the relation because $4 \geq 2\times 2$. $\left ( 2,1 \right )$ are also elements of the relation because $2 \geq 2\times 1$. But, $\left ( 4,1 \right )$ are not elements of the relation because $4$ is not greater than or equal to $2 * 1$.

Answer 1

$$x\ge2y\land y\ge2z\implies x\ge2y\land 2y\ge4z\implies x\ge 4z\implies x\ge2z.$$

Update: The last implication requires $z\ge0$.

Answer 2

What set is your relation defined on? If it's for positive numbers, then it is transitive, as Yves Daoust showed.

If the numbers can be negative, we could have: $$x = -3, y =-\frac{7}{4}, z = -1$$ Then $x = -3 \geq-3.5 =2y$, $y =-\frac{7}{4} \geq -2 = 2z$, but $x = -3 < -2 = 2z$.

Answer 3

Your relation is not transitive, since it fails to be transitive when you work with negative numbers. As an example, $$-4 \ge 2 \cdot (-2) \mbox{ and } -2 \ge 2 \cdot (-1)$$ but $$-4 \ge 2 \cdot (-1) \mbox{ is false}$$