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Which of the following statements about Regular Expression is/are incorrect? A. The union of two regular expressions is also a regular expression B. The concatenation of two regular expressions is also a regular expression C. The iteration of a regular expression is also a regular expression

Which of the following statements about Regular Expression is/are incorrect? A. The union of two regular expressions is also a regular expression B. The concatenation of two regular expressions is also a regular expression C. The iteration of a regular expression is also a regular expression Correct Answer Only C

The correct answer is option 1.

Concept:

Statement A: The union of two regular expressions is also a regular expression.

True,  The union of two regular expressions is closed it means the union of two regular expressions is gives a regular expression.  

Regular expression 1 ∪ Regular expression 2 = Regular expression.

Example:

RE1 = a(aa)* 

RE2 = (aa)*
So, L1 = {a, aaa, aaaaa,.....} 
and L2 ={ ε, aa, aaaa, aaaaaa,.......} 
L1 ∪ L2 = { ε, a, aa, aaa, aaaa, aaaaa, aaaaaa,.......}
RE = (L1 ∪ L2) = a* which is a regular expression.

Statement B: The concatenation of two regular expressions is also a regular expression.

True, The concatenation of two regular expressions is closed which means the concatenation of two regular expressions gives a regular expression.  

Regular expression 1 . Regular expression 2 = Regular expression.

RE1 = (0+1)*0 
RE2 = 01(0+1)*
L1 = {0, 00, 10, 000, 010, ......} 
L2 = {01, 010,011,.....} 
Then, L1.L2 = {001,0010,0011,0001,00010,00011,1001,10010,.............}  
L1.L2 =(0+1)*0 .  01(0+1)*
which is a regular expression.
Statement C: The iteration of a regular expression is also a regular expression. 
False, Iteration is the process of repeating a process in order to generate a series of outcomes. Each iteration of the process is a single repetition of the process, and the outcome of each iteration serves as the starting point for the next iteration. 
So iteration can perform many operations like Union, intersection, Subset, prefix, suffix, and many more.  So a subset of any regular expressions is not closed under regular grammar. Hence iteration is not closed under regular expressions.
Example:
(a+b)* are regular expressions it accepts all strings.
(anbn) is subset of above regular expressions and it is not a regular expression.
Hence the correct answer is Only C.

Additional Information Close Properties:

  Regular DCFL CFL CSL Recursive REL
Union Y N Y Y Y Y
Intersection Y N N Y Y Y
Complement Y Y N Y Y N
Difference Y N N Y Y N
Prefix Y Y Y Y Y Y
Suffix Y Y Y Y Y Y
Substring Y Y Y Y Y Y
Concatenation Y N Y Y Y Y
Reversal Y N Y Y Y Y
Kleen closure Y N Y Y Y Y
positive closure Y N Y Y Y Y
 subset N N N N N N
substitution Y N Y N N Y
Homomorphism Y N Y N N Y
Inverse Homomorphism Y Y Y Y Y Y

Y = Closed

N = Not closed

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