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

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

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)\}`.