Relation antisymmetry check

Question

Hello the question I am having trouble with is

Describe a binary relation on 1, 2, 3 that is reflexive and transitive, but not symmetric nor antisymmetric.

I Have the answer {(1,1),(1,2),(1,3),(2,2),(2,3),(3,3)}

is this right? i'm not sure if it is antisymmetric or not

Answer 1

The proposed relation is antisymmetric, meaning that all your non-reflexive relations are one-way (there are no cases of both $R(a,b)$ and $R(b,a)$ for $a\ne b$). You could just pick any one additional relation to remove the antisymmetry, because then you would have a case where both $(a,b)$ and $(b,a)$ are true. Preserving the transitivity might take a bit more work

Answer 2

Notice that symmetry and antisymmetry are properties that need to hold for all elements, so we just need to make sure there exists a counterexample to each in the relation.

The relation is not symmetric because you have included elements that break the symmetry, for example $(1,2)$ is in the relation, but $(2,1)$ is not.

However, your relation is antisymmetric because the only elements of the form $(a,b)$ and $(b,a)$ have that $a=b$; these are $(1,1)$, $(2,2)$, and $(3,3)$. So we need a counterexample that breaks the antisymmetry as well, for example by adding $(2,1)$ to the relation.