Number of relations which are reflexive but not symmetric

Question

I have following doubt. Pls have a look To find number of relations which are reflexive but not symmetric

———————————————————————————————

one way

$n(R-S)=n(R)-n(R \cap S) = 2^{(n^2-n)} - 1* 2^{n(n-1)/2} $———->(1 )

But if i think intuituively, In relations matrix,

reflexive means all diagonal elements present not symmetric means non-diagonal elements must be 1,0 or 0,1 or 0,0 for $a_{ij},a_{ji}$ It means non-diagonal elements have 3 possibibliees each It means $1*3^{n(n-1)/2}$ ——————> ( 2)

But (1) and (2) are different. I know i made some mistake but i am not able to identify on my own. Pls help

Answer 1

Your second way of counting is incorrect. Because not symmetric doesn't mean that all off-diagonal pairs have to be either $(1,0)$ or $(0,1)$ or $(0,0)$. Some pair could be $(1,1)$ and still the relation could be non-symmetric.

For example, the following matrix represents a relation that is reflexive and NOT symmetric. $$\begin{bmatrix}1&1&1\\0&1&1\\1&1&1\end{bmatrix}$$