Solution:

$\begin{aligned} &\text { Solution : Let } R^{*} \text { the transitive closure of } R \\\\ &\text { where } R=\{(1,2),(3,4),(4,5),(4,1),(1,1)\}\\\\ &\text { then } \mathrm{R}^{*}=\mathrm{R}^{\mathrm{R}} \mathrm{R}^{2} \cup \mathrm{R}^{3} \cup \ldots \cup \mathrm{R}^{\mathrm{k}} \cup \ldots\\\\ &R \cdot R=R^{2}=\{(3,5),(3,1),(4,2),(4,1),(1,1),(1,2)\}\\\\ &R \cdot(R \cdot R)=R^{3}=\{(3,2),(3,1),(4,1),(4,2),(1,1),(1,2)\}\\\\ &R^{4}=\{(3,1),(3,2),(4,1),(4,2),(1,1),(1,2)\}\\\\ &R^{5}=\{(3,1),(3,2),(4,1),(4,2),(1,1),(1,2)\}\\\\ &\mathrm{R}_{3}=\mathrm{R}_{4}=\mathrm{R}_{5}\\\\ &\mathrm{R}^{*}=\mathrm{R} \cup \mathrm{R}^{2} \cup \mathrm{R}^{3}\\\\ &R^{*}=\{(1,2),(3,4),(4,5),(4,1),(1,1),(3,5),(3,1),(4,2),(3,2)\} \\ \end{aligned}$