TOC Languages Difference

Question

Given two languages

L1 =$a^{*}(ab+a)$

L2=$a^{+}(ab+a)$

What is L1-L2?

Comments

I got {ab,a} .Please some one verify

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{a,ab} is correct

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yes correct, a + ab..

Answer 1

There is a lot that is common in between $L_1$ and $L_2$. In fact, out of observation it is clear that $L_2 \subset L_1$.

Again from observation, we can exactly identify the strings that are in $L_1$ but not $L_2$. These strings are the one which belongs to the regular expression (r.e)  $\epsilon(ab+a)$

Now, this r.e is obtained (as it might be obvious to you) by generating $\epsilon$ from $a^*$ in $L_1$. This is true because the only string that $a^*$ can generate but not $a^+$ is $\epsilon$ i.e empty string. Thus,

$$L_1 - L_2 = (ab+a)$$

 
HTH

Comments

answer is { a , ab}