$C o n s i d e r t h e D F A X g i v e n a b o v e o v e r \sum = 0, 1 . L e t L (X) b e t h e l a n g u a g e g e n e r a t e d b y t h e a b o v e D F A . W e a r e g i v e n f o l l o w i n g s t r i n g s . 1 . a^{32} {(b a)}^{128} 2 . a^{2017} b^{2018} a^{2019} 3 . {(a^{72} b^{14})}^{2} 4 . {(\in)}^{32} H o w m a n y o f t h e a b o v e s t r i n g s b e l o n g t o L (X)____.$

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Solution

The correct option is A 1
$L (X) = N u m b e r o f a' s a n d n u m b e r o f b' s i s m u l t i p l e o f 3 N o w l e t' s s e e t h e s t r i n g s, 1 : a^{32} {(b a)}^{128} n_{a} = 32 + 128 = 160 (W h i c h i s n o t d i v i s i b l e b y 3) S o t h i s s t r i n g d o e s n o t b e l o n g s t o L (X) . 2 : a^{2017} b^{2018} a^{2019} n_{a} = 2017 + 2019 = 4036 (W h i c h i s a l s o n o t d i v i s i b l e b y 3) S o t h i s s t r i n g d o e s n o t b e l o n g s t o L (X) . 3 : {(a^{72} b^{14})}^{2} n_{a} = 72 \times 2 = 144 (W h i c h i s d i v i s i b l e b y 3) n_{b} = 14 \times 2 = 28 (W h i c h i s n o t d i v i s i b l e b y 3) S o t h i s s t r i n g d o e s n o t b e l o n g s t o L (X) . 4 : \in^{32} n_{a} = 0 (W h i c h i s d i v i s i b l e b y 3) n_{b} = 0 (W h i c h i s d i v i s i b l e b y 3) S o t h i s s t r i n g b e l o n g s t o L (X) . S o o n l y 1 s t r i n g b e l o n g s t o t h e L (X) .$

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