in Theory of Computation
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in Theory of Computation
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L3’ = complement of L3 which is a$^{n}$ >=2

L$_{3}$.L$_{2}$.(L$_{1}$)*    here L1={phi}*=epsilon

so  L3’.L2.L1=a$^{n}$ >=2    --------------------eq1

now 2nd part  L1.L3’= {phi}.L3’=phi

 

so from eq 1 i think option D match..

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edited by
If it’s  L3 to the power 1 then answer C. If it’s the complement of L3 then the answer is D.

But it looks like 1. In a well-framed question, it will be mentioned.
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This could be solved using some basic Regular Expression Identities.

1)$\phi$*=$\varepsilon$

2)R.$\phi$=$\phi$.R=$\phi$

3)R.$\varepsilon$=$\varepsilon$.R=R

Where R is some Regular Expression.

Let's find L3'=Complement of L3=a*-L3

Since LE is defined over Alphabet With only symbol a.

L3'=a*-{a,$\varepsilon$}

L3'={$\varepsilon$,$a$,${a^2}$,$a^3$,$a^4$,.......} - {$a$,$\varepsilon$}

L3'={$a^2$,$a^3$ ,$a^4$,$a^5$,.......}

L3'={$a^n$/ $n$>=2}

Now,Solving the Expression

 L3'.L2.L1*+L1.L3'=L3'.$\varepsilon$.$\phi$*+$\phi$.L3'

                                 =L3'.$\varepsilon$.$\varepsilon$ +$\phi$(using $1$ and $2$)

                                   =L3'+$\phi$

                                    =L3'

                                    ={$ a^2$/$n$>=2}

​​​

Thus ,option D holds.

 

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Are you sure that 2nd property is correct ?

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Check it now.
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$L^{1}\ _{3}$  it’s not a complement, pls read carefully
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there nothing is mentioned whether it's complement or not

and if we don't write complement then ans will be c not d

So might be their printing mistake
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