Relational Sets for Reflexive, Symmetric, Anti-Symmetric and Transitive

Question

I just want to brush up on my understanding of Relations with Sets. Specifically with this set:

$\{ 1, 2, 3 \}$

I understand Reflexive, Symmetric, Anti-Symmetric and Transitive in theory. But if it's not too much trouble, I'd like some help producing the appropriate R (relation) sets with the set above. So in a nutshell:

Question: What's the Relation sets for Reflexive, Symmetric, Anti-Symmetric and Transitive on the following set?:

$\{ 1, 2, 3 \}$

Answer:

I've got some of them solved, but will complete my answer when I've figured out the rest.

Reflexive: $\{(1,1),(2,2),(3,3)\}$

Symmetric: $\{?\}$

Anti-Symmetric: $\{?\}$

Transitive: $\{(2,2),(2,3),(3,2),(3,3)\}$

Answer 1

First of all find $A\times A$ wherein $A=\{1,2,3\}$. So, we have $$A\times A=\{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)\}$$ Now try to choose some subset of above set as relations on $A$, since we know that every relation on $A$ has the form $R\subseteq A\times A$. For example $R_1=\{(1,1),(2,2),(3,3)\}$ is reflexive. $R_2=\{(1,1),(1,2),(2,1),(2,2)\}$ is reflexive, transitive and symmetric. Now try to find other relations according to what you have learnt about them.