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

Let $*$ be a binary operation defined on the set of rational numbers $Q$. Determine whether the binary operation defined by $a * b = a + ab$ for all $a, b \in Q$ is commutative.

On the set of positive rationals,a binary operation $*$ is defined by $a * b = \frac{2ab}{5}$. If $2 * x = 3^{-1}$,then $x = $

Let $^*$ be the binary operation on $N$ given by $a \,^*\, b = \text{L.C.M. of } a \text{ and } b$. Is $^*$ associative?

Determine whether or not each of the definitions of $*$ given below gives a binary operation. In the event that $*$ is not a binary operation,give justification for this. On $Z^{+}$,define $*$ by $a * b = a$.

Let $A = \{0, 1, 2, 3, 4, 5, 6\}$. If $a, b \in A$,and $a * b$ is defined as the remainder when $ab$ is divided by $7$,then find the inverse of $2$ under the operation $*$.