# Let A=\{a,\ b,\ c,\ d,\ e,\ f\}. Define the relation R=\{(a,a),(a,c),(b,d),(c,d),

Let $$\displaystyle{A}={\left\lbrace{a},\ {b},\ {c},\ {d},\ {e},\ {f}\right\rbrace}.$$ Define the relation $$\displaystyle{R}={\left\lbrace{\left({a},{a}\right)},{\left({a},{c}\right)},{\left({b},{d}\right)},{\left({c},{d}\right)},{\left({c},{a}\right)},{\left({c},{c}\right)},{\left({d},{d}\right)},{\left({e},{f}\right)},{\left({f},{e}\right)}\right\rbrace}$$ on A.
a) Find the smallest reflexive relation $$\displaystyle{R}_{{{1}}}$$ such that $$\displaystyle{R}\subset{R}_{{{1}}}$$.
b) Find the smallest symmetric relation $$\displaystyle{R}_{{{2}}}$$ such that $$\displaystyle{R}\subset{R}_{{{2}}}$$
c) Find the smallest transitive relation $$\displaystyle{R}_{{{3}}}$$ such that $$\displaystyle{R}\subset{R}_{{{3}}}$$.

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okomgcae

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