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

Show that $-a$ is the inverse of $a$ for the addition operation '$+$' on $R$ and $\frac{1}{a}$ is the inverse of $a \neq 0$ for the multiplication operation '$\times$' on $R$.

On the set $Z$ of all integers,the operation $*$ is defined by $a * b = a + b - 5$. If $2 * (x * 3) = 5$,then $x$ is equal to:

Let $^*$ be a binary operation on the set $Q$ of rational numbers defined as $a * b = (a - b)^2$. Determine whether the operation is commutative and associative.

Which of the following is false?

Let $P$ be the set of all subsets of a given set $X$. Show that $\cup: P \times P \rightarrow P$ given by $(A, B) \rightarrow A \cup B$ and $\cap: P \times P \rightarrow P$ given by $(A, B) \rightarrow A \cap B$ are binary operations on the set $P$.