Check whether the relation $R$ on $R$ defined as $R=\left\{(a, b):a\le b^3\right\}$ is reflexive, symmetric or transitive.

If $\mathrm{A}=\left\{1, 2, 3, 4\right\}$ define relations on $\mathrm{A}$ which have properties of being reflexive, transitive but not symmetric?

If $A=\{1,2,3,4\}$, define relations on $A$ which have properties of being symmetric but neither reflexive nor transitive

If $A=\{1,2,3,4\}$, define relations on $A$ which have properties of being reflexive, symmetric and transitive.

Let $\mathrm{R}$ be relation defined on the set of natural number $\mathrm{N}$ as follows: $\mathrm{R}=\left\{\right(x,y):x\in \mathrm{N},y\in \mathrm{N},2x+y=41\}$.

Find the domain and range of the relation $\mathrm{R}$. Also verify whether $\mathrm{R}$ is reflexive, symmetric and transitive.