EXERCISE 1.4

 Let $*$ be a binary operation on the set $Q$ of rational numbers as $a*b=a{b}^2.$ Find whether the binary operation is commutative or associative or both.

Let $*$ be a binary operation on the set $Q$ of rational numbers. Find which of the operations has identity.

Let $A=N\times N$ and $*$ be the binary operation on $A$ defined by $\left(a,b\right)*\left(c,d\right)=\left(a+c,b+d\right).$ Show that $*$ is commutative and associative. Find the identity element for $*$ on $A$, if any.

State whether the given statement is true or false. Justify.

For an arbitrary binary operation $*$ on a set $N, a*a=a \forall a\in N.$

State whether the given statement is true or false and Justify.
If $*$ is a commutative binary operation on $N,$ then $a*\left(b*c\right)=(c*b)*a$.

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