Anonym

2021-08-21

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

okomgcae

a) Obtain the reflexive closure that gives the smallest reflexive relation ${R}_{1}$ such that $R\subset {R}_{1}$
Thus, the reflexive closure is ${R}_{1}=\left\{\begin{array}{c}\left(a,a\right),\left(a,c\right),\left(b,d\right),\left(c,d\right),\left(c,a\right),\left(c,c\right),\\ \left(d,d\right),\left(e,f\right),\left(f,e\right),\left(b,b\right),\left(e,e\right),\left(f,f\right)\end{array}\right\}$
Therefore, the smallest reflexive relation ${R}_{1}$ such that $R\subset {R}_{1}$ is
${R}_{1}=\left\{\begin{array}{c}\left(a,a\right),\left(a,c\right),\left(b,d\right),\left(c,d\right),\left(c,a\right),\left(c,c\right),\\ \left(d,d\right),\left(e,f\right),\left(f,e\right),\left(b,b\right),\left(e,e\right),\left(f,f\right)\end{array}\right\}$
b) Obtain the symmetric closure that gives the smallest symmetric relation ${R}_{2}$ such that $R\subset {R}_{2}$
Thus, the symmetric closure is ${R}_{2}=\left\{\begin{array}{c}\left(a,a\right),\left(a,c\right),\left(b,d\right),\left(c,d\right),\\ \left(c,a\right),\left(c,c\right),\left(d,d\right),\left(e,f\right),\\ \left(f,e\right),\left(d,b\right),\left(d,c\right)\end{array}\right\}$
Therefore, the symmetric relation ${R}_{2}$ such that $R\subset {R}_{2}$ is
${R}_{2}=\left\{\begin{array}{c}\left(a,a\right),\left(a,c\right),\left(b,d\right),\left(c,d\right),\\ \left(c,a\right),\left(c,c\right),\left(d,d\right),\left(e,f\right),\\ \left(f,e\right),\left(d,b\right),\left(d,c\right)\end{array}\right\}$
c) Obtain the transitive closure that gives the smallest transitive relation ${R}_{3}$ such that $R\subset {R}_{3}$
Thus, the transitive closure is ${R}_{3}=\left\{\begin{array}{c}\left(a,a\right),\left(a,c\right),\left(b,d\right),\left(c,d\right),\left(c,a\right)\\ \left(c,c\right),\left(d,d\right),\left(e,f\right),\left(f,e\right),\left(a,d\right)\end{array}\right\}$
Therefore, the transitive relation ${R}_{3}$ such that $R\subset {R}_{3}$ is
${R}_{3}=\left\{\begin{array}{c}\left(a,a\right),\left(a,c\right),\left(b,d\right),\left(c,d\right),\left(c,a\right)\\ \left(c,c\right),\left(d,d\right),\left(e,f\right),\left(f,e\right),\left(a,d\right)\end{array}\right\}$

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