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 binary element for $*$ on A, if any.

Answer 1

We have,

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

Now,

$$\begin{gathered} \left(a,b\right)*\left(c,d\right)=\left(a+c,b+d\right)=\left(c+a,d+b\right) \\ \Rightarrow \left(a,b\right)*\left(c,d\right)=\left(c,d\right)*\left(a,b\right) \end{gathered}$$

So, $*$ is commutative.

Also,

$$\begin{gathered} \left(a,b\right)*\left[\left(c,d\right)*\left(e,f\right)\right]=\left(a,b\right)*\left(c+e,d+f\right) \\ =\left(a,b\right)*\left(c+e,d+f\right) \\ =\left(a+c+e,b+d+f\right) \\ =\left(a+c,b+d\right)*\left(e,f\right) \\ =\left[\left(a,b\right)*\left(c,d\right)\right]*\left(e,f\right) \\ \Rightarrow \left(a,b\right)*\left[\left(c,d\right)*\left(e,f\right)\right]=\left[\left(a,b\right)*\left(c,d\right)\right]*\left(e,f\right) \end{gathered}$$

So, $*$ is associative.

Let (x, y) be the binary element for $*$ on A.

$$\begin{gathered} \left(a,b\right)*\left(x,y\right)=\left(a,b\right)=\left(x,y\right)*\left(a,b\right) \\ \Rightarrow \left(a+x,b+y\right)=\left(a,b\right) \\ \Rightarrow a+x=a\mathrm{and}b+y=b \\ \Rightarrow x=0\mathrm{and}y=0 \end{gathered}$$

Hence, (0, 0) is the binary element for $*$ on A.