Relations counting

Question

I am learning relations ansni found question i cannot answer. How many relations talhat are non symetric and also non antisimetric are there?

I tried to draw a matrix but cannotsee the right answer

How can i find out? Thanks

Answer 1

Say we have a set $X$. A relation $R$ on $X$ is a subset $R \subseteq X \times X$. Now, assume we have a relation $R$ which we know is both symmetric and anti-symmetric and let $(x,y) \in R$ be an element in the relation. Since $R$ is symmetric, we must have $(y,x) \in R$, and since $R$ is anti-symmetric, this implies that $x = y$. Hence all elements of $R$ must be of the form $(x,x)$.

Conclusion: The only symmetric and anti-symmetric relations on $X$ are the identity relations $\{(y,y) \mid y \in Y \subseteq X\} \subseteq X \times X$ (and also vacuously the empty relation). For example, if $X$ has $1$ element $x_1$, then the only choices are $\emptyset$ and $\{(x_1, x_1)\}$, and if $X$ has $2$ elements $x_1, x_2$, then the choices are $\emptyset$, $\{(x_1,x_1)\}$, $\{(x_2,x_2\}$, and $\{(x_1,x_1),(x_2,x_2)\}$.

So if $X$ has $n$ elements, then the number of symmetric and anti-symmetric relations is...?