Question

Prove that for any$u,v,w\in B\left(n\right),$

1. $d\left(u,v\right)=d\left(v,u\right)$
   
2. $d\left(u,v\right)=0$if and only if $\mathit{u}\mathbf{=}\mathit{v}$

c.$d\left(u,v\right)=d\left(u+w,v+w\right)$

Short answer

1. It is proven that$d\left(u,v\right)=d\left(v,u\right)$
2. It is proven that $d\left(u,v\right)=0$if and only if $u=v$.
3. It is proven that$d\left(u,v\right)=d\left(u+w,v+w\right)$

Step-by-step solution

Lemma 16.1

Which says that,

1. $d\left(u,v\right)=Wt\left(u-v\right)$
   
2. $d\left(u,v\right)⩽d\left(u,w\right)+d\left(w,v\right)$
   

(a)Step 2: Proving that du,v=dv,u

By lemma 16.1 we have,

$\begin{array}{rcl}d\left(u,v\right)& =& Wt\left(u-v\right)\\ & =& Wt\left(u+\left(-v\right)\right)\\ & & \end{array}$

Now, we know that inverse element of $u\in B\left(n\right)$is u

So, above equation becomes,

$\begin{array}{rcl}d\left(u,v\right)& =& Wt\left(u-v\right)\\ & =& Wt\left(u+\left(-v\right)\right)\\ & =& Wt\left(-u+v\right)\\ & =& Wt\left(v-u\right)\\ & =& d\left(v,u\right)\\ & & \end{array}$

So,$d\left(u,v\right)=d\left(v,u\right)$

Hence proved.

(b)Step 3: Proving that du,v=0 if and only if u=v

We know that, $B\left(n\right)$is an abelian group.

Let,

$\begin{array}{rcl}d\left(u,v\right)& =& 0\\ & \iff & \forall i, u_i=v_i\\ & \iff & u=v\\ & & \end{array}$

So,$d\left(u,v\right)=0$if and only if $u=v$.

Hence proved.

(c)Step 4: Proving that du,v=du+w,v+w

By lemma 16.1 we have,

$\begin{array}{rcl}d\left(u,v\right)& =& Wt\left(u-v\right)\\ & =& Wt\left(u+w-w-v\right)\\ & =& Wt\left(\left(u+w\right)-\left(v+w\right)\right)\\ & =& d\left(u+w,v+w\right)\\ & & \end{array}$

So,$d\left(u,v\right)=d\left(u+w,v+w\right)$

Hence proved.