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

Anonym 2021-08-21 Answered
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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Expert Answer

okomgcae
Answered 2021-08-22 Author has 4043 answers

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

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