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Which one of the following languages over the alphabet $\{0,1\}$ is described by the regular expression: $(0+1)^*0(0+1)^*0(0+1)^*$?

1. The set of all strings containing the substring $\text{00}$
2. The set of all strings containing at most two $\text{0}$'s
3. The set of all strings containing at least two $\text{0}$'s
4. The set of all strings that begin and end with either $\text{0}$ or $\text{1}$
edited | 837 views

Counter example for other choices:

(A) 1010 is accepted which doesn't contain 00
(B) 000 is accepted
(D) 01 is not accepted
selected by

a) all string contain substring 00

regular expression = Anything 00 Anything  = (0+1)*00(0+1)*

b)all string contain atmost two 0's

regular expression = only 2 zeros  + only 1 zero  + No zero = Anthing 0 Anything 0 Anything + Anything 0 Anything  + Anything = 1*01*01* + 1*01* + 1*

here anything means that is made of only 1's bcoz we have to restrict 0 (only two or less) so anything cannot include 0.

c) all string contain atleast two 0's

regular expression = Anything 0 Anything 0 Anything = (0+1)*0(0+1)*0(0+1)*

d) all strings that begin and end with either 0 or 1

regular expression = either 0 or 1  Anything either 0 or 1 = (0+1)(0+1)*(0+1)

All are same for strings containing atleast 2 0's.
+1 vote
The regular expression has two 0′s surrounded by (0+1)* which means accepted strings must have at least 2 0′s.