Given the relation $\{(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)\}\,$ determine whether it is reflexive, transitive, symmetric, or anti-symmetric.
I found this set to be reflexive and symmetric. But not transitive and anti-symmetric.
Would it be correct to say that this set would be anti-symmetric if we remove either the element $(1,2)$ or $(2,1)$?
Also, the solution claims this set to be transitive. But I found it not to be so, due to the reasoning that $(2,3)$ and $(3,4)$ is not in the set.
Is my understanding of these ideas correct? Thank you.
Best Answer
It is reflexive if this is a relation over the set $\{1,2,3,4\}$, and yes, the relation is symmetric.
Yes, If we remove $(1,2)$ or $(2,1)$ then it is anti-symmetric.
The relation is transitive, we do not need $(2,3)$ and $(3,4)$ to be in the set. Especially there is no pairs in the relation $(2,x)$ and $(x,3)$, which is what we would need in order to force $(2,3)$ to be in the relation due to transitivity.