Is there relation that is symmetrical, transitive and non-reflexive? We must show that there exists some kind of $\alpha$ relation $\alpha ⊆ X \times X$ which has these conditions : if this relation is I and II type. I) symmetrical: if $∀x,x' ∈ X : (x, x') ∈ \alpha ⇒ (x', x) ∈ \alpha$ II) transitive: if $∀x, x', x'' ∈ \alpha : (x, x') ∈ \alpha (x',x'') ∈ \alpha ⇒ (x, x'') ∈ \alpha$ then there must be result of non-reflexivity for this $\alpha$ relation III) nonreflexive: if $∀x ∈ X : (x, x) $ $\notin \alpha$ Is there any kind of relation like this?

If $X=\\{x\\}$ or $X=\emptyset$, then it's easy to show that $\alpha=\emptyset$.

Suppose $X$ has more than one element. Take $x\
e y$ - elements of $X$. Suppose also that $(x,y)\in\alpha$, then $(y,x)\in\alpha$, hence $(x,x)\in\alpha$, which leads to contradiction.

Thus, $\alpha=\emptyset$.