Question

Is the L(r1) = L(r2) where r1 = (ab*+c)(λ+∅) and r2 = (a+c)(b*+∅)? what are the languages of L(r1), L(r2)? show the definition's using regular expression, assume Σ = (a,b,c)

Answer 1

Answer:   No, L(r1) is not equal to L(r2)

• L(r1) L(r2)
• Language accepted by r1 and Language accepted by r2 are not same.

• L(r1) = { c , a ,ab ,abb, abbb,...............}
• L(r2) = { c , a ,ab ,abb, abbb,.... ,cb,cbb,cbbb, ...........................}
• L(r1) L(r2)
• L(r1)   L(r2)

• r2 is accepting some more strings
• r1 is accepting less number of strings

Input Alphabet Given  Σ = (a,b,c)

Regular Expression -1

r1 = (ab*+c)(λ+∅)

r1 = (ab* λ + ab* ∅ + c λ + c ∅ )

r1 = (ab* λ + ab* ∅ + c λ + c ∅ )

r1 = (ab* λ + ∅ + c λ + ∅ )

r1 = (ab* λ +  c λ )

r1 = (ab* +  c ) // either single c or a followed by any number of b's (including zero b's)

L(r1) = { c , a ,ab ,abb, abbb,...............}

Regular Expression -2

r2 = (a+c)(b*+∅)

r2 = (a b*+a ∅ + c b*+c ∅)

r2 = (a b*+ ∅ + c b*+ ∅)

r2 = (a b*+ c b*)

// either  c followed by any number of b's (including zero b's) or a followed by any number of b's (including zero b's)

L(r2) = { c , a ,ab ,abb, abbb,.... ,cb,cbb,cbbb, ...........................}

Rule-1

• Anything x  ∅ =  ∅
• ab* ∅ =  ∅

Rule-2

• Anything x  λ  ( means epsilon ε) =  Anything
• ab* λ =  λ

Definition: A Regular Expression RE can be defined as;

If X is a RE denoting the language L(X) and Y is a RE denoting the language L(Y), then

1. X + Y is a RE   corresponding to the language L(X) ∪ L(Y)     
2. X . Y is a RE corresponding to the language L(X) . L(Y)
3. R* is a RE corresponding to the language L(R*)

Note:

• L (ε) = { ε } // epsilon ε is a RE indicates the language containing an empty string.

• L ( ∅ ) = { } // null symbol   ∅ is a RE denoting an empty language.

• L = {x} // x is a RE where