If $A = \{a, b, c\}$,then the number of binary operations on $A$ is

The inverse of $2010$ in the group $Q^{+}$ of all positive rational numbers under the binary operation $*$ defined by $a * b = \frac{ab}{2010}, \forall a, b \in Q^{+}$,is

Define a binary operation $^*$ on the set $\{0, 1, 2, 3, 4, 5\}$ as $a \,^* \, b = \begin{cases} a+b, & \text{If } a+b < 6 \\ a+b-6, & \text{If } a+b \geq 6 \end{cases}$. Show that $0$ is the identity for this operation and each element $a \neq 0$ of the set is invertible with $6-a$ being the inverse of $a$.

$A$ group $(G, *)$ has $10$ elements. The minimum number of elements of $G$,which are their own inverses is

Show that addition, subtraction, and multiplication are binary operations on $R$, but division is not a binary operation on $R$. Further, show that division is a binary operation on the set $R_*$ of nonzero real numbers.

Determine which of the following binary operations on the set $N$ are associative and which are commutative: $a \ast b = 1$ for all $a, b \in N$.