Question

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.

Answer 1

Answer:

The answer is $\{(1,1),(2,2),(3,3),(4,4),(1,2)(2,3)(3,4),(1,3),(1,4)\}$

Step-by-step explanation:

Remember that a reflexive relation $R\subset \mathcal{P}(X)$, where $\mathcal{P}(X)$ is the power set of $X$, is one which conteins the ordered pairs of the form $(a,a)$, for $a\in X$.

So, As the reflexive and transitive closure of $R$ (that we will denote by $\overline{R}$) is in particular reflexive, we must add to $R$  the elements $\{(1,1) , (2,2),(3,3),(4,4) \}$

A transitive relation $R$ is one in which if the pair $(a,b)$ and the pair $(b,c)$ are in there, then the pair $(a,c)$ must be there too.

So, to complete the relation $R$ to be reflexive and transitive we must add the pair $(1,3)$ (because $(1,2),(2,3)$ are in $R$), the pair $(2,4)$, and the pair $(1,4)$ because we added the pair $(2,4)$.

Therfore we have that $\overline{R}=\{(1,1),(2,2),(3,3),(4,4),(1,2)(2,3)(3,4),(1,3),(1,4)\}$.