I can't find any resource that would indicate whether symmetry and transitivity relations are possible or not with the existential quantifier or when quantifiers are combined.
I'm interested in the following types of relations: some-to-some, all-to-some-and some-to-none. If I'm not wrong the first two kinds are examples of surjective and the last one of non-surjective function.
Intuitively, it seems to me that only in the first case both symmetry and transitivity are possible. To be honest, I don't even know how to represent transitivity with the some-to-none relations.
Some to some:
Symmetry: $\exists$x$\exists$y(Rxy$\Rightarrow$Ryx)
Transitivity: $\exists$x$\exists$y$\exists$z(Rxy$\land$Ryx)$\Rightarrow$Rxz
All to some:
Symmetry: $\forall$x$\exists$y(Rxy$\Rightarrow$$\lnot$Ryx)
Transitivity: $\forall$x$\exists$y$\exists$z(Rxy$\land$$\lnot$Ryx)$\Rightarrow$$\lnot$Rxz
Some to none
Relation: $\exists$x(Sx$\land$$\forall$y(Sy$\Rightarrow$$\lnot$Ryx)
Symmetry: $\exists$x(Sx$\land$$\exists$y($\lnot$Sy)$\Rightarrow$(Rxy$\Rightarrow$$\lnot$Ryx)
Transitivity: ?