On the topic relations, is there any set in a relation R on $$ A = \{0,1,2,\{3\},4\} $$ which is reflexive, not symmetrical, not antisymmetrical and transitive? If a relation is transitive, shouldn't it also be at least antisymmetrical? For instance if the set is $$R = \{(0,1),(1,2),(0,2)\}$$ R should be transitive because $$(0,1) \land(1,2) \implies (0,2)$$ If R is transitive it should also be at least antisymmetrical, because $$(0,1)\land(1,0)\implies 1=0$$ The implication should be true because $$(1,0)\not\in R$$ $$ 1 \land0\implies0$$ $$ (0\implies 0) \Longleftrightarrow 1$$ is a true implication.
Is something wrong in my logical understanding on this topic? Am I missing something, I am not quite sure, if I really understood it correctly.
Try this $$R = \{(0,0),(1,1),(2,2), (\{3\},\{3\}), (4,4),(1,2),(1,\{3\}),(\{3\},1)\}$$