**Question:**

Let $A=\mathbf{N} \times \mathbf{N}$ 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 $\mathrm{A}$, if any.

**Solution:**

A = N × N

* is a binary operation on A and is defined by:

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

Let (*a, b*), (*c, d*) ∈ A

Then, *a, b, c, d* ∈ N

We have:

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

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

[Addition is commutative in the set of natural numbers]

∴(*a, b*) * (*c, d*) = (*c, d*) * (*a, b*)

Therefore, the operation * is commutative.

Now, let (*a, b*), (*c, d*), (*e, f*) ∈A

Then, *a, b, c, d, e*, *f *∈ N

We have:

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

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

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

Therefore, the operation * is associative.

An element $e=\left(e_{1}, e_{2}\right) \in$ A will be an identity element for the operation ${ }^{*}$ if

$a^{*} e=a=e^{*} a \forall a=\left(a_{1}, a_{2}\right) \in \mathrm{A}$, i.e., $\left(a_{1}+e_{1}, a_{2}+e_{2}\right)=\left(a_{1}, a_{2}\right)=\left(e_{1}+a_{1}, e_{2}+a_{2}\right)$, which is not true for any element in $\mathrm{A}$.

Therefore, the operation * does not have any identity element.

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