Let V be the set of ordered pairs $($a$,$ b$)$ of real numbers. Show that V is not a

vector space over R with addition and scalar multiplication defined by:

$($i$)($a$,$ b$)+($c$,$ d$)=($a $+$ d$,$ b $+$ c$)$ and k$($a$,$ b$)=($ka$,$ kb$),$

$($ii$)($a$,$ b$)+($c$,$ d$)=($a $+$ c$,$ b $+$ d$)$ and k$($a$,$ b$)=($a$,$ b$),$

$($iii$)($a$,$ b$)+($c$,$ d$)=(0,0)$ and k$($a$,$ b$)=($ka$,$ kb$),$

$($iv$)($a$,$ b$)+($c$,$ d$)=($ac$,$ bd$)$ and k$($a$,$ b$)=($ka$,$ kb$).$