Question

Let A = R $\times$ R and * be a 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. Also, find the inverse of every element (a, b) $ϵ$ A.

Answer 1

Given A = R $\times$ R and * is a binary operation on A defined by (a,b) * (c,d) = (a+c, b + d).

Commutativity : Let (a,b),(c,d) $ϵ$ A.

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

Hence, (a,b) * (c,d) = (c,d) * (a,b). $\therefore$ * is commutative.

Associativity : Let (a,b), (c,d), (e,f) = (a+c+c, b+d+f)=((a+c)+e,(b+d)+f)

$\Rightarrow =(a+(c+e),b+(d+f\left)\right)$

$\Rightarrow =(a,b)\ast (c+e,d+f)=(a,b)\ast \left[\right(c,d)\ast (e,f\left)\right]$

Hence, [(a,b) * (c,d)] * (e,f) = (a, b) * [(c,d) * (e,f)] ... * is associative.

Now, Let (x, y) be identity element for * on A.

Then (a, b) * (x,y) = (a,b) $\Rightarrow$ (a + x, b + y) = (a, b)

$\Rightarrow$ a + x = a, b + y = b $\Rightarrow$ x = 0, y = 0.

So, the identity element for the binary operation * is (0, 0) $\Rightarrow$ (a + e, b + f) = (0, 0)

$\Rightarrow$ a + e = 0, b + f = 0 $\Rightarrow$ e = -a, f = -b $\therefore$ inverse of (a, b) is (-a, -b).