Types of Relations

 1) 

2) Types of Relations

3) Identity Relation

4) Reflexive Relation 

Even if some elements are related to more than 1 element, it is compulsory for those elements to be related to themselves for the relation to be Reflexive. For example in the below example '1' R '1' and also '1' R '2' (here R means Relates to).

5) Total Number of Reflexive Relations of a Finite set Formula and Derivations

Here the n elements are compulsory but the rest of the elements (n² - n) have 2 possibilities, they can either come or not come. Therefore the 2 possibilities comes n² - n (remaining elements) times. 

Hence the formula 2 ^ n² - n .

6) Symmetric Relations

Note: null relations are also symmetric relations since the definition says if aRb then bRa should compulsorily be there but if aRb is not in the relation then bRa need not be in the relation too, and that is exactly the case with null relations, i.e., R = {}.

7) Total Number of Symmetric Relations of a Finite set Formula and Derivations

Here No elements are compulsory to be in the Relation for it to be symmetric (1, 1) , (2, 2) , (3, 3) has 2 possibilities, either they can come or not , so 2^3 possibilities for these 3 elements and the for the rest 6 elements they will be counted in pairs i.e., (1, 2) & (2, 1) are a pair and counted as one single entity. So 6/2 = 3 elements, and each of these 3 elements have 2 possibilities, they can come or not , therefore 2^3 possibilities.  So in total the number of symmetric Relation R that can be defined is 2^3 x 2^3 => 2^6

8) Total number of Reflexive and Symmetric Relations that can be defined on a Set A

For More Explanation go to this beautiful explanation

9) Transitive Relation

10) Equivalence Relations

for the transitive part : 

If (a, b) and (b, c) є R then (a, c) should also є R

so 1st triplets (a, b) = (1, 2)

                      (b, c) = (2, 1) 

                      (a, c) = (1, 1) and

2nd triplets    (a, b) = (2, 2)

                      (b, c) = (2, 1)

                      (a, c) = (2, 1) { notice that (b, c) & (a, c) is exactly same and that's absolutely fine as long as we are getting the triplets}.