a) Obtain the reflexive closure that gives the smallest reflexive relation \(\displaystyle{R}_{{{1}}}\) such that \(\displaystyle{R}\subset{R}_{{{1}}}\)

Thus, the reflexive closure is \(R_{1}=\left\{\begin{array}{c}(a,a),(a,c),(b,d),(c,d),(c,a),(c,c),\\ (d,d),(e,f),(f,e),(b,b),(e,e),(f,f)\end{array}\right\}\)

Therefore, the smallest reflexive relation \(\displaystyle{R}_{{{1}}}\) such that \(\displaystyle{R}\subset{R}_{{{1}}}\) is

\(R_{1}=\left\{\begin{array}{c}(a,a),(a,c),(b,d),(c,d),(c,a),(c,c),\\ (d,d),(e,f),(f,e),(b,b),(e,e),(f,f)\end{array}\right\}\)

b) Obtain the symmetric closure that gives the smallest symmetric relation \(\displaystyle{R}_{{{2}}}\) such that \(\displaystyle{R}\subset{R}_{{{2}}}\)

Thus, the symmetric closure is \(R_{2}=\left\{\begin{array}{c}(a,a),(a,c),(b,d),(c,d),\\ (c,a),(c,c),(d,d),(e,f), \\ (f,e),(d,b),(d,c)\end{array}\right\}\)

Therefore, the symmetric relation \(\displaystyle{R}_{{{2}}}\) such that \(\displaystyle{R}\subset{R}_{{{2}}}\) is

\(R_{2}=\left\{\begin{array}{c}(a,a),(a,c),(b,d),(c,d),\\ (c,a),(c,c),(d,d),(e,f), \\ (f,e),(d,b),(d,c)\end{array}\right\}\)

c) Obtain the transitive closure that gives the smallest transitive relation \(\displaystyle{R}_{{{3}}}\) such that \(\displaystyle{R}\subset{R}_{{{3}}}\)

Thus, the transitive closure is \(R_{3}=\left\{\begin{array}{c}(a,a),(a,c),(b,d),(c,d),(c,a)\\ (c,c),(d,d),(e,f),(f,e),(a,d)\end{array}\right\}\)

Therefore, the transitive relation \(\displaystyle{R}_{{{3}}}\) such that \(\displaystyle{R}\subset{R}_{{{3}}}\) is

\(R_{3}=\left\{\begin{array}{c}(a,a),(a,c),(b,d),(c,d),(c,a)\\ (c,c),(d,d),(e,f),(f,e),(a,d)\end{array}\right\}\)