Relations

Number of Relations

Number of Binary Relations on a single set $A$ (s.t. $|A|=n$ ). ^[https://en.wikipedia.org/wiki/Template:Number_of_relations] ^[https://math24.net/counting-relations.html].

Relation type	Number of relations on a Single Set of n-elemetns
Any ^[it’s the same as the number of functions from $A$ to $P(A)$ ]	$2^{n^2}$
Transitive
Reflexive or Irreflexive	$2^{n^2-n}$
Symmetric	$2^{\frac{n^2-1}{2}}$
Antisymetric	$2^n\cdot 3^{\frac{n^2-n}{2}}=2^n\sqrt{3^{n\left(n-1\right)}}.$
Asymmetric	$\sqrt{3^{n\left(n-1\right)}}$
Equivalence relation	Bell numbers
Total Order	$n!$

Number of any Relations from A to B

$2^{\mid A\mid \cdot \mid B\mid }=\mid P(A\times B)\mid$