How to prove transitive relations

Let $$R$$ be a binary relation on $$\mathbb{N}$$ defined by $$xRy$$ if and only if $$x − 2 ≤ y ≤ x + 2$$

How do you find if it is a transitive relation when there is only $$xRy$$? Isn't transitivity the relation between 2 conditions, for example $$xRy$$, $$yRz$$ therefore $$xRz$$ ?

• You have the right definition of transitivity. Now what do you think the answer to your question is? Just take three numbers in $x,y,z \in \mathbb{N}$ and suppose $xRy$ and $yRz$. Does $xRz$ or is there a counter-example? – Bill O'Haran Apr 29 at 15:29
• Also, for your next posts here please use MathJax. – Bill O'Haran Apr 29 at 15:34

You can obtain a counterexample for $$x=3, y=4, z=6$$.

So the relationship is not tranistive.

Bare in mind that: $$xRy: x − 2 ≤ y ≤ x + 2 \\ yRz: y− 2 ≤ z ≤ y + 2\\ xRz: x − 2 ≤ z ≤ x + 2$$

Transitivity of $$R$$ means that if we have, for some $$x, y, z \in \mathbb{N}$$, $$xRy$$ and $$yRz$$, then we automatically have $$xRz$$. You are right to say that transitivity is a ‘relation between two conditions’, but it isn’t a relation between two relations.

In this case, if $$R$$ is transitive then whenever $$x-2 \leq y \leq x+2$$ and $$y-2 \leq z \leq y+2$$, it should also be true that $$x-2 \leq z \leq x+2$$. Intuitively, this means that whenever $$y$$ is a distance of at most $$2$$ from $$x$$, and $$z$$ is a distance of at most $$2$$ from $$y$$, then $$z$$ must be a distance of at most $$2$$ from $$x$$. Let’s think. Does this seem reasonable? It does not, because clearly $$x$$ and $$y$$ can be separated by $$2$$, and then $$z$$ can be more than $$2$$ away from $$x$$, if it is greater than $$y$$.

A concrete counterexample would be: $$a=1,b=3,c=5$$. $$a$$ and $$b$$ are at most $$2$$ apart, and so are $$b$$ and $$c$$, but $$c$$ is more than $$2$$ away from $$a$$. So we have $$aRb$$ and $$bRc$$ but not $$aRc$$, so $$R$$ is not transitive. A single counterexample is all that is required to show that $$R$$ is not transitive, but if it was transitive, then we would need to prove this.

To prove that a relation is transitive, we need to start with the assumptions (that $$xRy$$ and $$yRz$$ for some arbitrary $$x,y,z$$ in the set) and deduce rigorously, perhaps using other known theorems, that $$xRz$$ must be true.