Definition of a transitive relation.

Question

For a given set X and a relation R, it is said to be transitive if, $(x,y)\in R \land (y,z)\in R \Rightarrow (x,z) \in R$. And there’s the case where for $X=\{1,2,3,4\}$, the relation $R =\{(1,2),(3,4)\}$ is transitive since $(x,y)\in R \land (y,z)\in R$ is always false. So, why define it that way instead of $(x,y)\in R \land (y,z)\in R \land (x,z) \in R$?, which is true, only if all of them are true, and if the first to cases are true, then is equivalent to the given definition. What makes the “then” definition stronger or better?

Answer 1

Because it doesn't mean the same thing. $$ ((x,y)\in R)\land ((y,z)\in R)\land ((x,z)\in R)$$ implies that $(x,y)\in R$ and $(y,z)\in R.$ But these don't need to hold. All transitivity says is if they hold, then $(x,z)\in R.$

The key problem with your reasoning is that you've assumed that in other examples where $((x,y)\in R)\land ((y,z)\in R)$ is not always false, it will always be true. This is not the case. It is possible (in fact quite likely) that it is true for some $(x,y,z)$ but not for others.