Kenneth Rosen Edition 6th Exercise 7.4 Example 2 (Page No. 489)

Question

Hi, following is a question from Kenneth H. Rosen :

Find the smallest relation containing the relation R ={(a, b) | a>b} that is both reflexive and symmetric?

The online solution given to the above problem is :

I couldn't understand the given solution. Isn't the question asking just for the closure(which should be reflexive and symmetric) of the given relation? Can someone please help me understand this question? 

Answer 1

Here the relation is asking for the reflexive relation and symmetric relation.
Here domain is not specified so let me take the positive natural number so :

Could you please suggest me some pair in the relation (a,b) where the relation is satisfied i.e

Is there any number that is greater than itself ? its null right?

Thus what is implied is that both these relation would be null ! because you wont be able to find the pair of number where the one number is greater than other and other way around as well?

I hope you get it?

Comments

Hi, I understand what you said but I am confused regarding the "smallest relation containing the relation" part. Doesn't it refer to the closure of this relation?

---

closure of a realtion is all about adding relative elements  to the set of relation so that the new relation holds the property of the repective relation .

For example if set of elements ={1,2} and R={(1,2),(1.1)} Then adding (2,2) would make the relation reflexive and similarly adding other pairs would make it symmetric and transitive .
Read this https://www.cs.rutgers.edu/~elgammal/classes/cs205/chapt74.pdf
or watch 19 lecture of Kamala Maam of discrete mathematics .
 

---

I think you mean R={(1,1),(2,2)(1,2)} to be as reflexive closure.

---

I already mentioned that after adding (2,2) it would be a reflexive relation.