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Let $r=1(1+0)^*, s=11^*0 \text{ and } t=1^*0 $ be three regular expressions. Which one of the following is true?

  1. $L(s) \subseteq L(r)$ and $L(s) \subseteq L(t)$

  2. $L(r) \subseteq L(s)$ and $L(s) \subseteq L(t)$

  3. $L(s) \subseteq L(t)$ and $L(s) \subseteq L(r)$

  4. $L(t) \subseteq L(s)$ and $L(s) \subseteq L(r)$

  5. None of the above

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L(r) = 1(1+0)* = {1,10,11,100,101,110,111, …..}

L(s) = $1^{+}$0 = {10,110,1110, ….}

L(t) = $1^{*}$0 = {0, 10,110,1110, ….}

From the above langages we can can write

L(s) ${\displaystyle \subset }$ L(t) ${\displaystyle \subset }$ L(r) 

Therefore  L(s) ${\displaystyle \subset }$  L(r)  and  L(s) ${\displaystyle \subset }$  L(t)  but not  L(s) ${\displaystyle \subseteq }$  L(r)  and  L(s) ${\displaystyle \subseteq }$  L(t)

Therefore the answer should be

E)None of the above

Correct me if i am wrong

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