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Find regular expression from the language

Write regular expression:

1. Find Regular expression for even length string where ∑ = {a, b}

Regular Expression 

2. Regular expression for all odd-length string where ∑ = {a, b}

Regular Expression 

Now if we combine all even and odd length string then we will get all string

Even length string and Odd length string = Set of all string

3. Regular Expression RE = (1 + 01)*. Find the language.

This RE is a set of all strings not containing two consecutive zero’s and ending with one.

4. Regular Expression RE = (λ + 0) (1 + 10)* Find the language.

Language of the Regular Expression

L = {set of all string not containing two consecutive zero}

5. Regular Expression = (1 + 011)* is set of all strings when every zero followed by at least two 1.

6. RE = (aa)* (bb)*b, (aa)* (bb)* b, $\mathbf{\{}{\mathbf{a}}^{\mathbf{n}}\mathbf{ }{\mathbf{b}}^{\mathbf{m}}\mathbf{,}\mathbf{ }\mathbf{n}\mathbf{ }\mathbf{\ge }\mathbf{ }\mathbf{0}\mathbf{,}\mathbf{ }\mathbf{m}\mathbf{ }\mathbf{\ge }\mathbf{ }\mathbf{0}\mathbf{\}}$set of all string where even no. of a’s followed by odd no. of b’s. 

7.  Language:

8. Language:

9. Regular Expression = (a + b)* (a + bb) So, language ={set of all string ending with a or bb}

10. Language L = {ω | |ω| ≥ 2} at least two length string ∑ = {a, b}

      String containing language L = { aa, ab, ba, bb, aaa, aab…..}

Then Regular Expression:

11.  Language L = {ω | |ω| = 2} exactly two length string ∑ = {a, b}

      Then Regular Expression RE = (a + b) (a + b) = ∑∑ 

12. Regular Expression RE = (a + b) a (a + b) (a + b)*. What is the language of this RE?

L = set of all strings having at least 3 lengths of string and the second symbol is a, from left.

13.  Regular Expression RE = (a + b)* a (a + b) (a + b). What is the language of this RE?

L = set of all strings having at least 3 lengths of string and the third symbol is a, from the right.

14.  Language L = {ω | |ω| = k} i.e. exactly k length string so, RE = ${\mathbf{(}\mathbf{a}\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{b}\mathbf{)}}^{\mathbf{k}}$

15. Language L = {ω | |ω| ≥ k} i.e. at least k length string then 

RE = ${\mathbf{(}\mathbf{a}\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{b}\mathbf{)}}^{\mathbf{*}}{(a + b)}^k$

16.  Language L = {ω | |ω| ≤ k} i.e. at most k length string then RE = ${\mathbf{(}\mathbf{\lambda }\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{a}\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{b}\mathbf{)}}^{\mathbf{k}}$

17.  Regular Expression RE = 0(0 + 1)*1 + 1(0 + 1)*0 then

Language L = set of all strings starting with 0 and ending with 1 or starting with 1 and ending with 0.

           Or,

Language L = set of all strings where first and last symbol are not the same