If the operation $ \oplus $ is defined by $ a \oplus b = a^{2} + b^{2} $ for all real numbers $ a $ and $ b $,then $ (2 \oplus 3) \oplus 4 = $

Consider a binary operation $*$ on $N$ defined as $a * b = a^{3} + b^{3}$. Choose the correct answer.

Given a non-empty set $X$,let $^*: P(X) \times P(X) \rightarrow P(X)$ be defined as $A \,^*\, B = (A - B) \cup (B - A)$,$\forall A, B \in P(X)$. Show that the empty set $\Phi$ is the identity for the operation $^*$ and all the elements $A$ of $P(X)$ are invertible with $A^{-1} = A$.

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.

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

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 $R$.