An example of a relation that is symmetric and anti-symmetric

[Magenta55]

Would this example be valid in satisfying a relation that is symmetric and anti-symmetric?

The relation R = {(1,1),(2,2)} on the set A = {1,2,3}

Also, I'm curious to know since relations can both be neither symmetric and anti-symmetric, would R = {(1,2),(2,1),(2,3)} be an example of such a relation? (I'm unsure if I'm using the definitions correctly).

 

[wabbit]

Would this example be valid in satisfying a relation that is symmetric and anti-symmetric?
The relation R = {(1,1),(2,2)} on the set A = {1,2,3}.

Yes, and that's essentially the only case : If R is both symmetric and antisymmetric then R must be the relation $\{(x,x),x \in B\}$ for some subset $B\subset A$.

Also, I'm curious to know since relations can both be neither symmetric and anti-symmetric, would R = {(1,2),(2,1),(2,3)} be an example of such a relation?

Yes. Symmetric or antisymmetric are special cases, most relations are neither (although a lot of useful/interesting relations are one or the other).