Is the set $A$ of anti-symmetric relations over $\mathbb{N}$ countable? I think this set is uncountable, but I don't really know how to prove this. Maybe somehow find an injection from $\mathcal{P}( \mathbb{N} \times  \mathbb{N}) \rightarrow A$?
For any $X\subseteq \mathbb N$, the relation $R_X = \{(x,x) | x \in X\}$ is anti-symmetric.
For any $S \subset \mathbb{N}$ take $\mathcal{R}_S = \bigcup_{s \in S} \{(s,s+1)\}$. It is clear that $\mathcal{R}$ is anti-symmetric, as it's a subset of anti-symmetric relation $\mathcal{R}=\{(x,y):x&lt;y\}$, and that this transformation is injective, so there are at least $\# \mathcal{P}(\mathbb{N})$ anti-symmetric relations on $\mathbb{N}$, thus their number is uncountable.