I am new to relations on sets and am trying to get my head around transitive relations.
I understand the definition of $(x,y) \in R, (y,z) \in R$ and $(x,z) \in R$ However what i am not sure about is if $x$ and $z$ need to be the same number.
For example if I have the following relation: $R = \{(1,2), (1,3), (1,4), (2,1), (2,3), (2,4), (3,1), (3,2), (3,4), (4,1), (4,1), (4,3)\}$
Answer 1: Can you this be transitive if you say $(2,1)$ is an element of $R$ and $(1,4)$ is an element of $R$ so is $(2,4)$ which makes this transitive.
Answer 2: Or does the definition mean that if $(2,1)$ is there $(1,2)$ is there but $(2,2)$ is not there and it is not transitive.
I am not sure of which one of the answers is correct and why?
The relation $R$ is transitive if
In order to prove a relation is transitive, you need to prove the previous definition holds for all $x,y,z$ in the set where the relation is defined. However, if you want to prove $R$ is not transitive, it suffices to exhibit some elements $x_0,y_0,z_0$ such that $(x_0,y_0),(y_0,z_0)\in R$ but $(x_0,z_0)\notin R$. In your example (answer two), your taking $x_0=z_0=2$ and $y_0=1$, it proves $R$ is not transitive.
The first answer you gave is wrong since you're proving the definition of transitive only for some $x,y,z$, not for all of them.