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

For each binary operation $^*$ defined below,determine whether $^*$ is commutative or associative. On $Q$,define $a ^* b = ab + 1$.

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.

In the group $(G, \times_{15})$,where $G = \{3, 6, 9, 12\}$ and $\times_{15}$ is multiplication modulo $15$,the identity element is

In $P(X)$,the power set of a non-empty set $X$,a binary operation $*$ is defined by $A * B = A \cup B, \forall A, B \in P(X)$. Under $*$,which of the following statements is true?

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