Question $4$: $18$ Marks

Let $A=\{1,2,3,4,5\}$ and

$R=\{(1,1),(1,3),(3,3),(3,5),(5,5)\}.$

Determine each of the following:

$(4\mathrm{.}1)$ The reflexive closure of $R.$

$(4\mathrm{.}2)$ The symmetric closure of $R.$

$(4\mathrm{.}3)$ The transitive closure of $R,$ using Warshall's algorithm.