Question

If * is a binary operation defined on A=N x N, by (a,b) * (c,d)=(a+c,b+d), prove that * is both commutative and associative. Find the identity if it exists.

Answer 1

Consider the problem

$A=N\times N$

$\left(a,b\right)\ast \left(c,d\right)=\left(a+b,b+d\right)$

For commutative

Let $\left(a,b\right)\in N\times Nand\left(c,d\right)\in N\times N$

then,

$\left(a,b\right)\ast \left(c,d\right)=\left(a+b,b+d\right)...\left(1\right)$

And

$\left(c,d\right)\ast \left(a,b\right)=\left(c+a,d+b\right)=\left(a+c,b+d\right)...\left(ii\right)$

From $\left(i\right)$ & $\left(ii\right)$

$\left(a,b\right)\ast \left(c,d\right)=\left(c,d\right)\ast \left(a,b\right)$

So, $\ast$ is commutative

For Associative

Let $\left(a,b\right),\left(c,d\right)\&\left(e,f\right)$ belongs to $A$

$\begin{array}{c}\left\{\left(a,b\right)\ast \left(c,d\right)\right\}\ast \left(e,f\right)=\left(a+c,b+d\right)\ast \left(e,f\right)\\ =\left(a+c+e,b+d+f\right)....\left(4\right)\end{array}$

Also

$\begin{array}{c}\left(a,b\right)\ast \left\{\left(c,d\right)\ast \left(e,f\right)\right\}=\left(a,b\right)\ast \left(c+e,d+f\right)\\ =\left(a+c+e,b+d+f\right)....\left(5\right)\end{array}$

From $\left(4\right)$ and $\left(5\right)$

$\left\{\left(a,b\right)\ast \left(c,d\right)\right\}\ast \left(e,f\right)=\left(a,b\right)\ast \left\{\left(c,d\right)\ast \left(e,f\right)\right\}$

So, $\ast$ is Associate

And Identity element does not exists.