Test whether the following relation is reflexive, symmetric or transitive on the set specified

$R=\left\{\right(m,n):m+n\text{ is not divisible by }3\} \mathrm{on} Z\text{ . }$

List the members of the equivalence relation defined by the partition $\{ \{1\}, \{2\}, \{3,4\} \}$ on $X=\left\{1,2,3,4\right\}$. Also find the equivalence classes of $1,2,3$ and $4$.

List the members of the equivalence relation defined by the partition $\{ \{1,2,3\}, \{4\} \}$ on $X=\left\{1,2,3,4\right\}$. Also find the equivalence classes of $1,2,3$ and $4$.

List the members of the equivalence relation defined by the partition $\left\{\right\{1,2,3,4\left\}\right\}$ on $X=\left\{1,2,3,4\right\}$. Also find the equivalence classes of $1,2,3$ and $4$.

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