Does $(f(0)=g(0)$ or $f(1)=g(1))$ define a transitive relation on function?

Question

I need is to check if a relation is an equivalence or not. I can see that it is reflexive and symmetric but I'm not able to find out if it is transitive.

The relation is defined on the set of all functions from $\mathbb{Z}$ to $\mathbb{Z}$ and the relation is :

$\left\{ (f,g) \bigm| f(0)=g(0) \text{ or } f(1)=g(1) \right\}$

Answer 1

Let R be the binary relation you mentioned. If:

• (f, g) ε R,
• (g, h) ε R,

then either f(0) = g(0) or f(1) = g(1).

Also either g(0) = h(0) or g(1) = h(1).

So it could be f(0) = g(0) and g(1) = h(1), while neither f(0) is equal to h(0) nor f(1) is equal to h(1).

Thus (f, h) doesn't belong to R, so R isn't transitive.