Chapter 8.3, Problem 43ES

To determine

(a)

To show that if $\left[\left(a,b\right)\right]=\left[\left(a^{\prime },b^{\prime }\right)\right]\text{ and}$ $\left[\left(c,d\right)\right]=\left[\left(c^{\prime },d^{\prime }\right)\right],\text{ then}$

$\left[\left(ad+bc,bd\right)\right]=\left[\left(a^{\prime }{d}^{\prime }+b^{\prime }{c}^{\prime },b^{\prime }{d}^{\prime }\right)\right].$

To determine

(b)

To prove:

If $\left[\left(a,b\right)\right]=\left[\left(a\text{'},b\text{'}\right)\right]$ and $\left[\left(c,d\right)\right]=\left[\left(c\text{'},d\text{'}\right)\right],\text{ then }\left[\left(ac,bd\right)\right]=\left[\left(a\text{'}c\text{'},b\text{'}d\text{'}\right)\right].$

To determine

(c)

To show that $\left[\left(0,1\right)\right]$ is an identity element for addition. That is, show that for any $\left(a,b\right)\in A,$

$\left[\left(a,b\right)\right]+\left[\left(0,1\right)\right]=\left[\left(0,1\right)\right]+\left[\left(a,b\right)\right]=\left[\left(a,b\right)\right].$

To determine

(d)

An identity element for multiplication for the given operation. That means, find $\left(i,f\right)$ in A so that for all $\left(a,b\right)$ in A. $\left[\left(a,b\right)\right]\cdot \left[\left(i,j\right)\right]=\left[\left(i,j\right)\right]\cdot \left[\left(a,b\right)\right]=\left[\left(a,b\right)\right].$

To determine

(e)

To prove:

To show that $\left[\left(-a,b\right)\right]$ is an inverse for $\left[\left(a,b\right)\right]$ for addition if $\left(a,b\right)\in A,$. That means, to show that $\left[\left(-a,b\right)\right]+\left[\left(a,b\right)\right]=\left[\left(a,b\right)\right]+\left[\left(-a,b\right)\right]=\left[\left(0,1\right)\right].$

To determine

(f)

An inverse for $\left[\left(a,b\right)\right]$ for multiplication for any $\left(a,b\right)\in A\text{ with }a\ne 0$. That means, to find $\left(c,d\right)$ in A so that $\left[\left(a,b\right)\right]\cdot \left[\left(c,d\right)\right]=\left[\left(c,d\right)\right]-\left[\left(a,b\right)\right]=\left[\left(i,j\right)\right],$ where $\left[\left(i,j\right)\right]$ is the identity element you found in part (d).