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

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.

Determine whether or not the definition of $*$ given below gives a binary operation. In the event that $*$ is not a binary operation,give justification for this. On $R$,define $*$ by $a * b = ab^2$.

On the set of all natural numbers $N$,which one of the following $*$ is a binary operation?

Consider the binary operation $\wedge$ on the set $\{1, 2, 3, 4, 5\}$ defined by $a \wedge b = \min\{a, b\}$. Write the operation table of the operation $\wedge$.

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