et

$\backslash$Sigma $3=$

$\%$& $0$

$0$

$0$

$\text{'}$

$,$

& $0$

$0$

$1$

$\text{'}$

$,$

& $0$

$1$

$0$

$\text{'}$

$,...,$ & $1$

$1$

$1$

$\text{'}(.$

$\backslash$Sigma $3$ contains all size $3$ columns of $0$s and $1$s$.$ A string of symbols in $\backslash$Sigma $3$ gives three

rows of $0$s and $1$s$.$ Consider each row to be a binary number and let

B $=\{$w in $\backslash$Sigma $$

$3|$ the bottom row of w is the sum of the top two rows$\}.$

For example,

& $0$

$0$

$1$

$\text{'}$ & $1$

$0$

$0$

$\text{'}$ & $1$

$1$

$0$

$\text{'}$

in B$,$ but & $0$

$0$

$1$

$\text{'}$ & $1$

$0$

$1$

$\text{'}$

$$ in B$.$

Show that B is regular. $($Hint: Working with BR is easier. You may assume the

result claimed in Problem $1\mathrm{.}31.)$