Non Diagonal Pair and Couting the Relations(2)

3. Symmetric relation:-

Diagonal elements have two choices either they want to be the part or they don't. But for remaining elements, if (a,b) is present then (b, a) must be present.

So  Non - Diagonal elements are chosen in this way to participate in a relation.

the total number of Non-Diagonal element are : n^{2} -n 

 and the total number of Non-Diagonal Pairs are:-  (n^{2}-n)/2

 each pair have two choices: either they want to be part of relation or they don't.

Why we are making pairs:  By making pair we are assuring either both (a,b) and (b, a) is present or both are absent.  Two elements of type (a,b) and (b,a) forms a non-diagonal pair.

 

So total number of Symmetric relations is:-  2^{(n^{2}-n)/2}. 2^{^{n}}

 

Example:- 

let us say A={1,2,3} then 

and A relation R from A to A  is shown below:

R= {(1,1) ,(1,2) , (1,3)

        (2,1) ,(2,2) , (2,3)

        (3,1) ,(3,2) ,  (3,3) }

Now If you see ,clearly the bold letters represent a diagonal and and

Total number of diagonal elements is               3

Total number of Non - Diagonal element is :-  6 

Total number of Non - Diagonal pairs :-    6/2=3 and they are like this:-

1^{st} pair ={(1,2)(2,1)}

2^{nd} pair ={(1,3)(3,1)}

3^{rd} pair ={(2,3)(3,2)}

 

 So total number of Symmetric relations  = 2^{3}. 2^{3} = 64

 

 

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