Statements regarding relations in R

Question

Suppose $\rho$ is a relation on $R$. I want to verify whether the following statements are true. Looks simple but proving them seems to be difficult for me.

• $\rho\circ\rho$ is a subset of $\rho$
• $\rho\circ\rho=\rho$ implies $\rho=i_{D(\rho)}$ ($D(\rho)$ being the domain of $\rho$)
• $\rho=\rho^{-1}$ implies $\rho=i_{D(\rho)}$

I believe the second point is false considering the counter example constant function but I need help with the other statements.

Answer 1

All of them are false.

For the first, let $\rho = \{(a,b),\,(b,c)\}$

For the second, let $\rho = \{(1,1),\,(3,3),\,(1,3),\,(3,1)\}$

For the third, let $\rho = \{(0,1),\,(1,0)\}$

Answer 2

For an arbitrary binary relation, the first is false: if $\rho=\{(0,1), (2,0)\}$, then $(2,1)\in\rho\circ\rho$.

Regarding the second, I don't quite see what $i_{D(\rho)}$ is... actually, what is $D(\rho)$ itself — a subset of $\mathbb R$, or of $\mathbb R^2$)?

Assuming $D(\rho)$ is the subset of $\mathbb R$ consisting of all the first components of $\rho$, then consider$\rho=\{(0,1)\}\cup\{(x,x)\}_{x\in\mathbb{R}}$. You do have $\rho\circ\rho=\rho$.