Question

Let $A=Q\times Q$ and let* be a binary operation on $A$ defined by $(a,b)\ast (c,d)=(ac,b+ad)$ for $(a,b),(c,d)\in A.$ Determine, whether ${}^{\ast }$ is commutative and associative. Then, with respect to ${}^{\ast }$ on $A$.Find the identity element of $A$.

Answer 1

(i) For any (a, b), (c, d), (e, f)$\in A$, we have ${}^{\prime }{\ast }^{\prime }$

$\left\{\right(a,b)\ast (c,d\left)\right\}\ast [e,d]$

$=(ac,b+ad)\ast (e,f)$

$=(ace,b+ad+acf)$

and

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

$=(a,b)\ast (ce,d+cf)$

$=(ace,b+ad+acf)$

So, $\left\{\right(a,b)\ast (c,d\left)\right\}\ast (e,f)=(a,b)\ast \left\{\right(c,d)\ast (e,f\left)\right\}$ for all a, b, c, d, $\in Q\times Q=A$

${\Rightarrow }^{\prime }{\ast }^{\prime }$ is associative on A

(ii) Let (x, y) e identify element of A

$(a,b)\ast (x,y)=(a,b)=(x,y)\ast (a,b)$ for all $(a,b)\in A$

$\Rightarrow (ax,b+ay)=(a,b)=(xa,y+bx)$ for a, b$\in$Q

$\Rightarrow (ax,b+ay)=(a,b)$ and $(a,b)$

$\Rightarrow (xa,y+bx)$ for all $a,b\in Q$

$\Rightarrow ax=a$ & $b+ay=b$ for all $a,b\in Q$

$xa=a$, $y+bx=b$ for all $a,b\in Q$

$\Rightarrow x=1,y=0$

$(1,0)\in Q\times Q=A$

So, $(1,0)$ is identify element of A.