Determine the Lebesgue measure of the subset of [$0,1$] whose elements have neither first nor second digit in their decimal expansion equal to "$0$"
My workings for this problem are as follows:
If $x$ ${\in\mathbb [0,1]}$ has the form $x=0.0...$, then $x$ ${\in\mathbb [\frac{0}{10} ,1)}$$=:$ $A$
If $x$ ${\in\mathbb [0,1]}$ has the form $x=0.x_10...$, then $x$ ${\in\mathbb [\frac{0}{100} ,\frac{1}{10})\ \bigcup [\frac{10}{100} ,\frac{2}{10})\ \bigcup ...\ \bigcup [\frac{80}{100} ,\frac{9}{10})\ \bigcup [\frac{90}{100} ,1)\ }$$=:$ $B$
To calculate the Lebesgue measure of $A$ and $B$ I then used the formula $\lambda(A \bigcup B)= \lambda(A) +\lambda(B)-\lambda(A∩B)$
i.e. $1+10(\frac{10}{100})-\frac{10}{100}=\frac{19}{10}=$ $\lambda(A \bigcup B)$
$[0,1]=\lambda(A \bigcup B)\bigcup (A \bigcup B)^c$
So $\lambda(A \bigcup B)^c = 1-\lambda(A \bigcup B) = 1-\frac{19}{10}=\frac{-9}{10}$
This is the method my professor used worked out a similar problem in the notes, however he did not get a negative value for his answer. I'm doubting my solution is correct since it is a negative value for a measure. Any help or push in the right direction would be much appreciated!