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 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.

Which one of the following is not true?

Determine whether or not each of the definitions of $*$ given below gives a binary operation. In the event that $*$ is not a binary operation,give justification for this. On $Z^{+}$,define $*$ by $a * b = a$.

Show that $+: R \times R \rightarrow R$ and $\times: R \times R \rightarrow R$ are commutative binary operations,but $-: R \times R \rightarrow R$ and $\div: R_* \times R_* \rightarrow R_*$ are not commutative.

Let $*$ be a binary operation defined on $Q$. Determine whether the binary operation defined by $a * b = a - b$ for all $a, b \in Q$ is commutative.