let A and B are set such that |A| =n and |B| = m
then |A*B| = m*n
Relation: It is a subset of cartesian product.
Since already we know |A*B| = m*n and each element in the cartesian product have two choices: either they want to be the part of the relation of they don't want.
2 choices for m*n elements so total no. relations possible = 2m*n
we have already discussed the types of relations. So let's know some more details:
1. Number of Reflexive relations:-
let |A| = n
Reflexive relations are always defined on the same set :
|A*A| = n2
if there are n elements in the set then there will be n reflexive pairs like (1,1), (2,2) ... (n,n)
So all these pairs must be present in reflexive pairs so out of n2 elements n has one choice as they must be the part of relation but remaining n2 -n has two choice
So total number of reflexive relations=
2. The number of Ir-Reflexive relations:
Again those reflexive pairs have only one choice they must not be present so: one choice for 'n' elements and two choices for n2 -n elements.
total number of Ir- reflexive relations=