# Number of Quasi reflexive and coreflexive relations on a set with $n$ elements

Given a binary relation $$R$$ over a set $$A$$ with $$n$$ elements,then :

$$R$$ is coreflexive if:

$$\forall a,b \in A:aRb \implies a=b$$

$$R$$ is Quasi-reflexive if: $$\forall a,b \in A:aRb \implies aRa \;\;\;\text{and}\;\;\; bRb$$

• How many coreflexive and Quasi-reflexive relations exist on $$A$$?

Define $$A:=\left\{a_i \mid i \in I\right\}\tag{\left|I\right|=n,n \in \mathbb N}$$

Based on the definition $$(a_i,a_j)\in R$$ if $$a_i=a_j$$ .

On the other hand for each such $$i$$,either $$(a_i,a_i)\in R$$ or $$(a_i,a_i) ∉ R$$,so

Easily follows the number of coreflexive relation on $$A$$ is $$2^n$$.

For the other question let $$i.

If $$(a_i,a_j)\in R$$,then so are $$(a_i,a_i)$$ and $$(a_j,a_j)$$,also for each such $$i$$ either $$(a_j,a_i)\in R$$ or $$(a_j,a_i)∉ R$$,if $$(a_i,a_j)∉ R$$,then the only uncounted case appears when $$(a_i,a_j)\in R$$,from here it's seen that for all such $$i$$ there are 3 distinct Quasi-reflexive relations on $$A$$

Now we are left with the number of such $$i$$ which is $$\sum_{k=1}^{n-1}k=\frac{n\left(n-1\right)}{2}$$

So the number of Quasi-reflexive relations with $$i \ne j$$ is $$3^{\large\frac{n\left(n-1\right)}{2}}\tag{I}$$

Also for equal indexes either $$(a_i,a_i) \in R$$ or $$(a_i,a_i) ∉ R$$,the number of such $$i$$ is $$n$$,follows the number of such Quasi-relations is $$2^n\tag{II}$$

Summing $$(\text{I})$$ and $$(\text{II})$$ gives thew total number of Quasi-reflexive relations on $$A$$ for $$n\ge2$$.

But I'm not sure if the results are true,can someone check them?