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Let $r, s, t$ be regular expressions. Which of the following identities is correct?

1. $(r + s)^* = r^*s^*$
2. $r(s + t) = rs + t$
3. $(r + s)^* = r^* + s^*$
4. $(rs + r)^* r = r (sr + r)^*$
5. $(r^*s)^* = (rs)^*$

edited | 762 views
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This question was repeated in $\mathbf{2015}$ as well.
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1. $(r + s)^* = r^*s^*$                    LHS can generate '$sr$' but RHS not
2. $r(s + t) = rs + t$                 LHS can generate '$rt$' but RHS not
3. $(r + s)^* = r^* + s^*$              LHS can generate '$sr$' but RHS not
4. $(rs + r)^* r = r (sr + r)^*$    They are equivalent
5. $(r^*s)^* = (rs)^*$                      LHS can generate '$rrrs$' but RHS not

So option D is correct answer.
by Boss (16.5k points)
edited by
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WHY NOT C option .....correct ...explain

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i think i have given the reason
'sr' can be generated from LHS but not RHS
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Hint for Option D -

$(rs + r)r= (rsr + rr) = r(sr+r)$ (post multiply and then pre common )

can u take it from here ?
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Hi I have a doubt here.

(r+s)∗= can generate sr

Can you explain how (r∗s∗)* generates sr

In other words how they are equivalent

(r+s)∗=(r∗s∗)*
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Option D is a right choice.

by Loyal (5.2k points)
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perfect!
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whats X ...?
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