Ir-Reflexive and Reflexive

Let A= {1,2,3,4,5,6,........ n}

1. The cardinality of smallest reflexive relations.

2. The cardinality of smallest Ir-reflexive relation.

3. The cardinality of largest Ir reflexive relation.

4.  Choose the correct:

R1= {(1,1) , (1,2), (2,4)} is reflexive 

R2= {(1,1) , (2,5)} is  ir-reflexive.

Answer

Solutions: 

1. we already know that for a relation to be reflexive "All the diagonal elements must be present " and number of diagonal elements is n So The cardinality of smallest reflexive relations is =

Diagonal elements are : (1,1) (2,2) (3,3) (4,4) .................(n,n)

 

2. Irreflexive means there should be no diagonal elements. 

So  The cardinality of the smallest Ir-reflexive relation is 0 = Ø or  {   }

 

3. It should not contain any diagonal elements so remaining elements are  n^{2} -n . This is max. Cardinality.

 

4. for R1: No, because pairs like (2,2) ,(3,3)................(n,n) is not present. 

     for R2: No, because there is a diagonal element (1,1). for irreflixive, there should not be any diagonal elements .

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