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 = $

Let $^*$ be the binary operation on $N$ given by $a \,^*\, b = \text{L.C.M. of } a \text{ and } b$. Is $^*$ associative?

Let $*$ be a binary operation defined on the set of rational numbers $Q$. Determine whether the binary operation defined by $a * b = a^{2} + b^{2}$ for all $a, b \in Q$ is commutative.

Is $^*$ defined on the set $\{1, 2, 3, 4, 5\}$ by $a \,^*\, b = \text{L.C.M. of } a \text{ and } b$ a binary operation? Justify your answer.

Let $A = N \times N$ and $^*$ be the binary operation on $A$ defined by $(a, b) \,^*\, (c, d) = (a + c, b + d)$. Determine whether the operation $^*$ is commutative,associative,and has an identity element.

Let $*^{\prime}$ be the binary operation on the set $\{1, 2, 3, 4, 5\}$ defined by $a *^{\prime} b = \text{H.C.F. of } a \text{ and } b$. Is the operation $*^{\prime}$ same as the operation $*$ defined in Exercise $4$ above? Justify your answer.