Let $n$ be a fixed positive integer. Define a relation $\mathrm{R}$ in $\mathrm{Z}$ as follows $\forall a, b\in \mathrm{Z}:a\mathrm{R}b$ if and only if $a-b$ is divisible by $n$. Show that $\mathrm{R}$ is an equivalence relation.

If $\mathrm{A}=\{1,2,3,4\}$, define relation on $\mathrm{A}$ which have properties of being: reflexive, transitive but not symmetric.

If $\mathrm{A}=\{1,2,3,4\}$, define relations on $\mathrm{A}$ which have properties of being:

symmetric but neither reflexive nor transitive

If $\mathrm{A}=\{1,2,3,4\}$, define relations on $\mathrm{A}$ 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.

Given, $\mathrm{A}=\{2,3,4\}, \mathrm{B}=\{2,5,6,7\}$. Construct an example of an injective mapping from $\mathrm{A}$ to $\mathrm{B}$.

Given, $\mathrm{A}=\{2,3,4\}, \mathrm{B}=\{2,5,6,7\}$. Construct an example of a mapping from $\mathrm{A}$ to $\mathrm{B}$ which is not injective.

Given, $\mathrm{A}=\{2,3,4\}, \mathrm{B}=\{2,5,6,7\}$. Construct an example of a mapping from $\mathrm{B}$ to $\mathrm{A}$.

Give an example of a mapping which is one-one but not onto.