# How to construct relations on the following domain?

Construct the following relations on the domain {1,2,3}.

• A relation that is transitive and symmetric, but not reflexive or antisymmetric.

How would I construct the above relation?

I have attempted similar problems with the two being below.

• A relation that is both symmetric and antisymmetric.

{(1,2),(2,1),(3,2),(2,3)}

• A relation that is both transitive and not symmetric.

{(1,2),(2,3),(1,3)}

• Note that your example about a relation that is both symmetric and anti-symmetric is wrong. Why? Commented Mar 26, 2019 at 4:24
• Would this be correct for the relation with symmetric and anti-symmetric? {(1,2),(2,1),(2,3)} Commented Mar 26, 2019 at 4:56
• To be antisymmetric, it must be that if $(1,2)$ is in the relation, then you cannot have $(2,1)$ in the relation. Commented Mar 26, 2019 at 16:41

Here’s a hint.

Assume that $$(1,2)$$ was part of this relation you want. Since the relation is symmetric, ...

we must have $$(2,1)$$ in our relation.

Furthermore, since the relation is transitive, that means ...

$$(1,2)$$ and $$(2,1)$$, along with transitivity, imply that $$(1,1)$$ is also in our relation.

Now, what can be said about reflexivity?