Let $A=N\times N$. * defined on $A$ by $(a, b)*(c, d)=(a+c, b+d)$. Show that $A$ is closed for $*$.

Let $A=N\times N$. Define $*$ on $A$ by $(a,b)*(c,a)=(a+c,b+d)$. Show that $*$ is commutative,

Let $A=N\times N$. Define $*$ on $A$ by $(a,b)*(c,a)=(a+c,b+d)$. Show that $*$ is associative.

Let $A=N\times N$. Define $*$ on $A$ by $(a,b)*(c,a)=(a+c,b+d)$. Show that identity element does not exist in $A$.

Let $A=(1,-1,i,-i)$ be the set of four $4$th roots of unity. Prepare the composition table for multiplication on $A$ and show that $A$ is closed for multiplication.

Let $A=(1,-1, i,-i)$ be the set of four $4^{th}$ roots of unity. Prepare the composition table for multiplication on $A$ and show that multiplication is associative on $A$.

Let $A=(1,-1\mathit{,}i,-i)$ be the set of four $4^{th}$ roots of unity. Prepare the composition table for multiplication on $A$ and show that multiplication is commutative on $A$.

Let $A=(1,-1\mathit{,}i,-i)$ be the set of $4^{th}$ roots of unity. Prepare the composition table for multiplication on $A$ and show that $1$ is the multiplicative identity.

Let $A=(1,-1\mathit{,}i,-i)$ be the set of four $4^{th}$ roots of unity. Prepare the composition table for multiplication on $A$ and show that every element in $A$ has its multiplicative inverse.