(Possibly Imperfect) Proof #3

You should work on this problem individually (no partners) and submit your answers as a PDF file to Gradescope.
Read the below proof carefully and then answer the questions at the end.

Problem: Let A = {w | w contains at least one 0 and at least one 1}. Either prove that A is a context-free language or prove that A is not a context-free language.

Claim: A is a not a context-free language.

Proof:

Denote the alphabet of A as Σ = {0, 1}. Consider the regular expression R given by:

R = 0 0* 1 Σ* ∪ 1 1* 0 Σ*

Since L(R) = A, the language A can be generated by a regular expression. Any language that can be generated by a regular expression is a regular language. Thus, A is a regular language and A is not a context-free language. ■


Questions

  1. What type of proof are you providing feedback for (e.g., proof by contradiction, proof using a definition, proof by induction, etc.)?
  2. Do you find the proof presented in a way that is clear or confusing to understand?
  3. What is one thing that you like about this proof and/or that this proof does well?
  4. What is one thing that could be improved in this proof?
  5. The proof is not entirely correct. What makes this proof incorrect? Please explain your reasoning carefully here, and point out succinctly what might have gone wrong in the proof.