Let A$=$N × N and $*$ be the binary operation on A defined by (a, b) $*$ (c, d ) $=$ (a + c, b + d )

Show that $*$ is commutative and associative. Find the identity element for $*$ on A, if any.

A $=$ N $\times$ N and $*$ is a binary operation defined on A.

For commutativity

(a, b) $*$ (c, d)$=$ (a + c, b + d) $=$ (c + a, d + b) $=$ (c, d )$*$ (a, b)

The operation $*$ is commutative.

For associativity

[(a, b) $*$ (c, d)]$*$ (e, f )$=$ (a + c, b + d)$*$ (e,f)
$=$ (a + c + e, b + d + f )

Also, (a, b) $*$ [(c, d)$*$ (e, f )] $=$ (a, b) $*$ (c + e, d + f)

$=$ (a + c + e, b + d + f )

$\therefore [(a, b) *$ (c, d)] $*$ (e, f ) $=$ (a, b) $*$ [(c, d) $*$ (e, f)]

Hence, the operation $*$ is associative.

For identity

Let identity function be (e, f )
$\therefore$ (a, b) $*$ (e, f ) $=$ (a, b)

(a + e, b + f) $=$ (a, b) $\Rightarrow$ a + e $=$ a, b + f $=$ b

$\Rightarrow$ e $=$ 0, f $=$ 0. But, 0 $\notin$ N

Hence, identity element does not exist.