I would like to understand the transitive property in relations...I just cant get it in my brain. I mean the definition is crystal clear. However I still struggle. For example:
Given the set $A=\{0,1,2\}$ the $R=\{(0,0),(0,1),(1,1),(2,2)\}$
According to the definition if $(a,b) \in R$ and $(b,c) \in R \to (a,c)\in R$
So $0\sim0$ and $0\sim 1$ then I need $(0,1)$ again? I makes no sense for me, I mean the numbers are the same..I mean is $a=b$ also possible?
If $\sim$ is a transitive relation on $A$ then if $(a,b)\in R$ and $(b,c)\in R$ then $(a,c)\in R$. You have $0\sim 0$ and $0\sim 1$ which implies $0\sim 1$. Therefore $(0,1)\in R$, which is something that's true. Recall that$\{a,a,a,a,a\}=\{a\}$ so you don't need $(0,1)$ to be repeated in A, you already have it.