L(r) = { w E { a , b }* : w starts and ends with the same symbol }

r = ?

Pick all correct answers.

Question

L(r) = { w E { a , b }* : w starts and ends with the same symbol }

r = ?

Pick all correct answers.

BlackTom AI · Accepted Answer

Strings must either be a single symbol a or b, or have length ≥2 with the first and last symbol equal; the regular expression a(a+b)*a + b(a+b)*b + a + b exactly captures these cases.

1. Relevant concepts
- Alphabet Σ = {a,b}. 
- (a+b)* denotes any string (possibly empty) over Σ. 
- A regular expression concatenation like aR a denotes strings beginning with a and ending with a, with R as the middle part. 
- The empty string (λ) is included by (a+b)* but inclusion depends on whether the whole pattern allows λ.

2. Step-by-step reasoning
- Consider strings of length 1: "a" and "b" start and end with the same symbol; they must be included explicitly.
- Consider strings of length ≥2:
  - Those that start and end with 'a' are described by a (a+b)* a. Here (a+b)* can be empty, so aa is included.
  - Those that start and end with 'b' are described by b (a+b)* b, similarly including bb.
- Combine these with union: a(a+b)*a + b(a+b)*b, and add singletons + a + b.

3. Intermediate validation
- a(a+b)*a produces aa, aba, aaaa, etc. (middle can be ε or any string).
- b(a+b)*b produces bb, bab, bbbb, etc.
- Adding + a + b includes single-letter strings. No term allows λ, so ε is excluded (correct, since ε has no first/last symbol).

4. Why the given correct answer is correct
- It covers all and only strings whose first and last symbols are equal: singletons (a,b) and all longer strings captured by the two patterns. No extraneous strings (like those starting with a and ending with b) are generated.

5. Why other options are incorrect
- Option with "+ λ": incorrectly includes the empty string (ε), which should be excluded.
- (a+b)(a+b)*(a+b) variants do not enforce first and last symbols being equal; they allow any start and end, so include many invalid strings.
- "None is correct" is false because the second option is correct.

6. Conclusion
Therefore the correct regular expression is r = a(a+b)*a + b(a+b)*b + a + b, which matches the provided answer.

L(r) = { w E { a , b }* : w starts and ends with the same symbol } r = ? Pick all correct answers. 多项选择题

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