Recall the definitions first.
- $t\colon M\to M$ is a function if $t\subseteq M\times M$ such that for every $R\in M$ there is a unique ordered pair $\langle R,R'\rangle\in t$. We often denote $R'$ as $t(R)$.
- A function $t$ is called injective if for every $R,S$ in the domain of $t$ such that $R\neq S$ we have that $t(R)\neq t(S)$.
- A function $t\colon A\to B$ is called surjective if for every $C\in B$ there is at least one $R\in A$ such that $t(A)=B$.
Now when will $t$ be injective? If whenever you are given two different relations their transitive closure is different. Recall that if $R$ is transitive then $t(R)=R$, and so if $S$ is a non-transitive relation we have that $t(S)=t(t(S))$ but $t(S)\neq S$. So in order to find a counterexample for injectivity we need to point out at least one non-transitive relation.
Similarly when will $t$ be surjective? If every relation is a transitive closure of some other relation. Again a non-transitive relation will be a counterexample to this property.
Can you find a non-transitive relation?