So I have this exercise in my discrete math course that I don't understand:
Put $A = \{1,2,3\}$ and form the relation R on A by putting $R = \{(1,1),(2,2),(1,2),(2,3),(3,1)\}.$
Investigate if $R$ is: reflexive, symmetric, antisymmetric. If the relation has a property, give proof for it and if the relation doesn't have the property, prove it.
My attempt:
Reflexive: Yes, since $\{1,2,3\} = \{1,2,3\}.$
Symmetric: No, because in R every sub-pair of elements has max 2 elements and thus doesn't contain $\{1,2,3\}.$
Anti-symmetric: No ... but don't know why.
There is no hindsight but I know I'm wrong, can someone please help me out?
Thanks in advance,
Not reflexive as $(3,3) \notin R$.
Not symmetric as $(1,2) \in R$ but $(2,1) \notin R$.
The relation is anti symmetric.
Not transitive as $(1,2),(2,3) \in R$ but $(1,3) \notin R$.