Question

Let the number of strings in the following
$L_1=0^{\ast }{1}^{\ast },L_2=1^{\ast }{0}^{\ast },L_3=(0+1{)}^{\ast }\text{and}{L}_4=0^{\ast }{1}^{\ast }{0}^{\ast }.$
language are_____
$L=(L_1\cap L_2)-(L_3\cap L_4)$

• A. 0

Answer 1

The correct option is A 0
Given $L_1=0^{\ast }{1}^{\ast },L_2=1^{\ast }{0}^{\ast },L_3=(0+1{)}^{\ast }\text{and}{L}_4=0^{\ast }{1}^{\ast }{0}^{\ast }$

$L_1\text{can be simplified as,}{L}_1=0^{\ast }+1^{\ast }+0^{\ast }{1}^{\ast }$

$\text{Similarly}{L}_2=1^{\ast }+0^{\ast }+1^{\ast }{0}^{\ast }$

$\text{Therefore,}{L}_{\cap }{L}_2=0^{\ast }+1^{\ast }$

$\text{Since}{L}_3=(0+1{)}^{\ast }\text{is a superset of}{L}_4,\text{So we can say that}{L}_3\cap L_4=L_4$

$\text{So in this case}{L}_3\cap L_4=0^{\ast }{1}^{\ast }{0}^{\ast }$

$\text{Hence,}L=(L_1\cap L_2)-(L_3\cap L_4)$

$=(0^{\ast }+1^{\ast })-\left(0^{\ast }{1}^{\ast }{0}^{\ast }\right)=0$

$\text{So the numuber of strings in the language}L=0.$