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Let $S$ and $T$ be languages over $\Sigma=\{a,b\}$ represented by the regular expressions $(a+b^*)^*$ and $(a+b)^*$, respectively. Which of the following is true?

1. $S \subset T$
2. $T \subset S$
3. $S = T$
4. $S \cap T = \phi$

C is correct

S=(a+b*)*=(a*(b*)*)*=(a*b*)*

T=(a+b)*=(a*b*)*     so S=T
can anyone explain option D?
If two sets are equal then how intersection can be null??

A={1,2,3} B ={1,2,3} so when A=B then their intersection can't be null.

Some of the regular expression equivalent to (a+b)*  which might help

(a+b)* =(a*+b*)* = (a*b*)* = (a*+b)* =(a+b*)* =a*(ba*)* =b*(ab*)*

Typo. Comma {a,b}

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Correct Option: C

$S=T$. Both generates all strings over $\Sigma$.

by

Can anyone explain in more detail? I am confused that regular expression S will generate more number of b's as compared to expression T. If they need to be equal they both should always generate same set of symbol but number of b's generated by expression S can be more.

Is this solution is an identity of regular expression? or we need to evaluate set of string and make the comparison.

see the above comment both are equal

option C is ok but what about option D?
Both $S$ and $T$ can generate all strings over $\Sigma$ so their intersection is also = all strings over $\Sigma$  (not $\phi$)
Okay got it
If option A would have been S⊆ T then it would be an ambiguous question.
C is correct

S=(a+b*)*=(a*(b*)*)*=(a*b*)*

T=(a+b)*=(a*b*)*     so S=T

Bt string abbbabbb is not accepted by T which is accepted by S...then how S=T?Please correct me if I m wrong...
T = (a+b)*  this language is set of all possible strings it will have all strings belongs to alphabet  Σ={a.b}

(a+b)(a+b)(a+b)(a+b)(a+b)(a+b)(a+b)(a+b)(a+b)

now take a from first(a+b) then b from second(a+b)and so on... in this way ur string will be accepted by T..
Ohh...now I understand Thank You

### 1 comment

Can we use regular expression identities over alphabet a and b?

Here P and Q are regular expression

(P+Q)*=(P*+Q)*

P+Q=Q+P

So, (P+Q)*=(Q+P)*=(Q+P*)*

Also , from regular expression identities.I'm unable to obtain (P+Q*)*