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.
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 = n
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 . 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 .