The number of different equivalence relations on a set $S$ having $2$ elements is $2$.

State whether the statement is true/false:

The number of binary relations that can be defined on a set with $3$ elements is $2^3$.

State whether the following is true/false:

Union of two equivalence relations on a set is again an equivalence relation on that set.

If $A=\left\{1,2,3\right\}$, then the relation $R=\left\{\right(1,1),(2,2),(3,3),(1,3),(3,2),(1,2\left)\right\}$ on $A$ is

If $A = \left\{1, 2, 3\right\}$ then the relation $R=\left\{(1,1),(2,2),(1,2),(2,1),(1,3)\right\}$) on the set $A$ is

If $A = \left\{a, b, c\right\}$, then the relation $R=\left\{(a,a),(b,b),(c,c),(b,c),(a,b)\right\}$ on the set $A$ is

The relation $R=\left\{(1,1),(2,2),(4,4),(1,2),(1,3),(2,3)\right\}$ on the set $A=\left\{1,2,3,4\right\}$ is

Let $R*=R\backslash \left\{0\right\}$ and $\mathrm{\rho }$ be a relation on $R* \times R*$ defined by $(x, y) \mathrm{\rho }(z, t)\iff xt=yz$ for any $(x, y), (z, t) \in R* \times R*$. Then the relation $\mathrm{\rho }$ is

The void relation on a non-empty set $A$ is