Question

Complete the proof of Lemma 10.28 by showing that

(a)Addition of equivalence classes is associative.

(b)Multiplication of equivalence classes is associative.

(c)Multiplication of equivalence classes is commutative.

Short answer

It is proved that

(a) Addition in equivalence classes is associative.

(b) Multiplication in equivalence classes is associative.

(c) Multiplication in equivalence classes is commutative.

Step-by-step solution

Prove that addition in equivalence classes is associative

Now, by theorem 10.26 addition in equivalence class is defined as:

$\left[a,b\right]+\left[c,d\right]=\left[ad+bc,bd\right]$

Now, let$\left[a,b\right],\left[c,d\right],\left[e,f\right]\in F$ .

Simplify LHS as:

$\begin{array}{rcl}& \Rightarrow & \left\{\left[a,b\right]+\left[c,d\right]\right\}+\left[e,f\right]=\left\{\left[ad+bc,bd\right]\right\}+\left[e,f\right]\\ & =& \left[adf+bcf+bde,bdf\right]\end{array}$

Simplify RHS as:

$\begin{array}{rcl}& \Rightarrow & \left[a,b\right]+\left\{\left[c,d\right]+\left[e,f\right]\right\}=\left[a,b\right]+\left[cf+de,df\right]\\ & =& \left[adf+bcf+bde,bdf\right]\end{array}$

Therefore, LHS=RHS.

$\left\{\left[a,b\right]+\left[c,d\right]\right\}+\left[e,f\right]=\left[a,b\right]+\left\{\left[c,d\right]+\left[e,f\right]\right\}$

Hence, addition in equivalence classes is associative.

Prove that multiplication in equivalence classes is associative

Now, by theorem 10.26, multiplication in equivalence class is defined as:

$\left[a,b\right]\left[c,d\right]=\left[ac,bd\right]$

Now, let $\left[a,b\right],\left[c,d\right],\left[e,f\right]\in F$.

Now, simplify LHS= $\left\{\left[a,b\right]\left[c,d\right]\right\}\left[e,f\right]$as:

$\begin{array}{rcl}\left\{\left[ac,bd\right]\right\}\left[e,f\right]& =& \left[ace,bdf\right]\\ & =& \left[a,b\right]\left[ce,df\right]\\ & =& \left[a,b\right]\left\{\left[c,d\right]\left[e,f\right]\right\}\\ & =& RHS\end{array}$

Hence, $\left\{\left[a,b\right]\left[c,d\right]\right\}\left[e,f\right]=\left[a,b\right]\left\{\left[c,d\right]\left[e,f\right]\right\}$

Therefore, multiplication in equivalence classes is associative.

Prove that multiplication in equivalence classes is associative

Now, by theorem 10.26, multiplication in equivalence class is defined as:

$\left[a,b\right]\left[c,d\right]=\left[ac,bd\right]$

Now, let$\left[a,b\right],\left[c,d\right]\in F$ .

$\begin{array}{rcl}\left[a,b\right]\left[c,d\right]& =& \left[ac,bd\right]\\ & =& \left[ca,db\right][\therefore Risintegraldomain]\\ & =& \left[c,d\right]\left[a,b\right]\end{array}$

$\Rightarrow \left[a,b\right]\left[c,d\right]=\left[c,d\right]\left[a,b\right]$

Hence, multiplication in equivalence classes is associative.