e) Given the following: Let X = {1, 2, 3, 4) and a relation R on X as R= {(1,2), (2,3), (3,4)}. Find the reflexive and transitive closure of R.
Accepted Solution
A:
Answer: The answer is [tex]\{(1,1),(2,2),(3,3),(4,4),(1,2)(2,3)(3,4),(1,3),(1,4)\}[/tex]Step-by-step explanation:Remember that a reflexive relation [tex]R\subset \mathcal{P}(X)[/tex], where [tex]\mathcal{P}(X)[/tex] is the power set of [tex]X[/tex], is one which conteins the ordered pairs of the form [tex](a,a)[/tex], for [tex]a\in X[/tex].So, As the reflexive and transitive closure of [tex]R[/tex] (that we will denote by [tex]\overline{R}[/tex]) is in particular reflexive, we must add to [tex]R[/tex] the elements [tex]\{(1,1) , (2,2),(3,3),(4,4) \}[/tex]A transitive relation [tex]R[/tex] is one in which if the pair [tex](a,b)[/tex] and the pair [tex](b,c)[/tex] are in there, then the pair [tex](a,c)[/tex] must be there too.So, to complete the relation [tex]R[/tex] to be reflexive and transitive we must add the pair [tex](1,3)[/tex] (because [tex](1,2),(2,3)[/tex] are in [tex]R[/tex]), the pair [tex](2,4)[/tex], and the pair [tex](1,4)[/tex] because we added the pair [tex](2,4)[/tex].Therfore we have that [tex]\overline{R}=\{(1,1),(2,2),(3,3),(4,4),(1,2)(2,3)(3,4),(1,3),(1,4)\}[/tex].