Why Is $x \ne y$ Not Transitive on the Set of All integers?

Question

I know this is a pretty simple question, but I'm just not getting the textbook... I'm taking a basic CS course and on one of the problems (not an assigned homework problem, just one I'm practicing on), it says that

on the set of all integers, the relation $x \ne$ y is symmetric but not transitive where $(x,y) \in \Bbb R$.

I understand why it's symmetric, but why is it not transitive? I'm thinking it has something to do with the definition of transitive including ALL $a,b,c \in A$, but I'm not sure. I understand transitive in the context of a finite set, but something about applying it to all integers is throwing me off.

Answer 1

You don't need infinite sets at all - a set with two elements is enough.

Suppose $a\not=b$. Then $a\not=b$ and $b\not=a$. But $a=a$!

Indeed, "$\not=$" isn't transitive on any set with at least two elements.

Answer 2

For example $1\neq 2$ and $2\neq 1$, but is $1\neq 1$?