In the group $G=\{0, 1, 2, 3, 4, 5\}$ under addition modulo $6$,$(2 +_{6} 3^{-1} +_{6} 4)^{-1}$ is equal to

In the set of integers $(Z, *)$,if $a * b = a + b - n, \forall a, b \in Z$,where $n$ is a fixed integer,then the inverse of $(-n)$ is:

In a group $(G, *)$,for some element $a$ of $G$,if $a^{2}=e$,where $e$ is the identity element,then

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

Given a non-empty set $X$,consider the binary operation $^*: P(X) \times P(X) \rightarrow P(X)$ defined by $A \,^*\, B = A \cap B$ for all $A, B \in P(X)$,where $P(X)$ is the power set of $X$. Show that $X$ is the identity element for this operation and $X$ is the only invertible element in $P(X)$ with respect to the operation.

Let $*$ be a binary operation defined on $R$ by $a * b = \frac{a+b}{4}$ for all $a, b \in R$. Then the operation $*$ is: