# 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.

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.

• No, the given relation is transitive, since for every $aRb$ and $bRc$, we have $aRc$ in the relation, where $a,b,c\in\{1,2,3,4\}$. – Prasun Biswas Apr 1 '15 at 9:41
• Transitivity would require $(2,3)$ to be in the set only if there were some $n\in\{1,2,3,4\}$ such that $(2,n)$ and $(n,3)$ were in the set. There is no such $n$, however, so transitivity says nothing about the pair $(2,3)$. – Brian M. Scott Apr 1 '15 at 9:42
• Just to clarify, on which set is the relation made? Is it $R:A\mapsto A$ where $A=\{1,2,3,4\}$ – Prasun Biswas Apr 1 '15 at 9:43
• @PrasunBiswas Yes, sorry I forgot to include it. – user265675 Apr 1 '15 at 9:44
• @Meryll Note that that detail is of paramount importance. – Git Gud Apr 1 '15 at 9:47

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.
• So following that, even if we remove $(2,1)$ from the set, it can still remain transitive? Is this correct? – user265675 Apr 1 '15 at 9:53
• Yes. Since for the remaining "weird" relation $(1,2)$ we do not have anything on the form $(2,x)$ except for $(2,2)$ which is left in the relation. Note though that if we remove (1,1) from our original relation, you do not only lose reflexivity, but also transitivity since $(1,2)$ and $(2,1)$ implies $(1,1)$. – Ove Ahlman Apr 1 '15 at 10:58