Why can't a set {(1,1)} be an equivalence relation of set A={1,2,3}?

Question

I know that {(1,1),(2,2),(3,3)} is an equivalence relation of the set A. But I am not sure why can't the set{(1,1)} be an equivalence relation?
I think it is because the equivalnce relation is specified for all (a,a) which belong to set A so elemnt 1,2 and 3 should all belong to the set and not any one or two(2 or 3 for example) only.Is this correct? (I am only taking pairs with same elements here on purpose to clearly elucidate my point).
What if I have to create an equivalence class such that neither a nor b is equal to 3 or 3 is not a factor of a (in (a,b) where both a and b belong to set A).
Will such an equivalence relation ({(1,1),(2,2),(1,2),(2,1)}) exist or not?

Answer 1

Not only is it not reflexive, as mentioned in the comments, your proposed equivalence relation doesn't partition the set $A$. That means that each element of the set needs to be in a unique equivalence class. $2$ and $3$ from $A$ are left out, so it's not an equivalence relation.