Why is the transitivity fulfilled for two elements? Let $A=\\{a,b\\}$ be a set with two elements $a\neq b$. Define all order relations on $A\times A=\\{(a,a),(a,b),(b,a),(b,b)\\}$. Why is the transitivity for two elements fulfilled?

To prove that a relation $\preceq$ on a set $A = \\{a,b\\}$ is an order relation, you just have to prove that $\preceq$ is reflexive and antisymmetric, as transitivity follows from that:

If $x,y,z \in A$ are arbitrary with $x \preceq y$ and $y \preceq z$ than (as $A$ only has two elements) two of $x,y$ and $z$ must be equal. If $x = z$ than $x \preceq z=x$ by reflexivity, if $x=y$, we have $x =y\preceq z$, if $y=z$ than $x \preceq y=z$ by assumption.