Show that if $R$ is an equivalence relation on $X$ then $\mathrm{domain} R=\mathrm{range} R=X$

Give an example of a relation which is reflexive, symmetric but not transitive.

Give an example of a relation which is reflexive, transitive but not symmetric

Give an example of a relation which is symmetric, transitive but not reflexive.

Give an example of a relation which is reflexive but neither symmetric nor transitive

Give an example of a relation which is transitive but neither reflexive nor symmetric

Give an example of a relation which is an empty relation.

Give an example of a relation which is a universal relation.

Let $R$ be a relation on $X$, If $R$ is symmetric then $xRy\Rightarrow yRx$. If it is also transitive then $x\mathrm{R}y\text{ and }y\mathrm{R}x\Rightarrow x\mathrm{Rx}$. So whenever a relation is symmetric and transitive then it is also reflexive. What is wrong in this argument?