$4\mathrm{.}7$ Let $F_q$ be a finite field and let $S$ be a subset of $F_q^n.$

$($i$)$ Show that and $F_q^{n}S{F}_q{F}_q($:$S$:$)S=\{101,111,010\},q=2S=\{1020,0201,2001\},q=3S=\{00101,10001,11011\},q=2$