L(r) = { w E { a , b }* : w ends with ab OR starts with ab }

r = ?

Pick all correct answers:

Question

L(r) = { w E { a , b }* : w ends with ab OR starts with ab }

r = ?

Pick all correct answers:

BlackTom AI · Accepted Answer

The language L consists of all strings over {a,b} that either start with the substring "ab" or end with the substring "ab".

1) Relevant knowledge points and concepts
- Regular expression operators: concatenation, Kleene star (*) (zero or more repetitions), and union (+).
- (a+b)* denotes any string (including the empty string) over {a,b}.
- For a fixed prefix p, the expression p (a+b)* denotes all strings that begin with p. For a fixed suffix s, (a+b)* s denotes all strings that end with s.
- Union of two regexes denotes strings that satisfy either property (or both).

2) Step-by-step reasoning
- To capture "starts with ab": use ab concatenated with any continuation, ab (a+b)*. This includes the string "ab" itself because (a+b)* can be empty.
- To capture "ends with ab": use any prefix followed by ab, (a+b)* ab. This also includes "ab" (where the prefix is empty).
- Combine these with union to get all strings that either start with ab or end with ab: ab (a+b)* + (a+b)* ab.

3) Intermediate observations and simplifications
- Note that some other forms in the options attempt to force both start and end (e.g., ab (a+b)* ab) or add redundant stars (ab (a+b)* (a+b)* ab). Since (a+b)*(a+b)* = (a+b)*, ab (a+b)* (a+b)* ab simplifies to ab (a+b)* ab, which requires both a starting and ending "ab" — not equivalent to "start OR end".
- Overlap: strings that both start and end with "ab" are included because union accepts strings satisfying either or both properties.

4) Why the correct answer is correct
- ab (a+b)* + (a+b)* ab exactly matches the English description: the first term covers all strings starting with "ab", the second covers all strings ending with "ab". Their union covers the required set and nothing more.

5) Why the other options are incorrect
- "None is correct.": false because ab (a+b)* + (a+b)* ab correctly describes the language.
- "r = ab b* ab": describes strings of the form "ab" followed by some number of b's and then "ab" — this forces a specific middle pattern and requires both start and end; it excludes many valid strings.
- "r = ab (a+b)* (a+b)* ab" and "r = ab (a+b)* ab": both require the string to start with "ab" AND end with "ab", i.e., intersection rather than union; they miss strings that only start or only end with "ab".

Conclusion: The correct regular expression is ab (a+b)* + (a+b)* ab, which matches the provided answer.

L(r) = { w E { a , b }* : w ends with ab OR starts with ab } r = ? Pick all correct answers: Multiple choice

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