Let l1 l2 and l3 be languages over some alphabet in each

Question

Problem

Let L₁, L₂, and L₃ be languages over some alphabet ∑. In each case below, two languages are given. Say what the relationship is between them. (Are they always equal? If not, is one always a subset of the other?) Give reasons for your answers, including counterexamples if appropriate.

Let a be any element of A. Let b be any element of A for which aRb. Then since R is symmetric, bRa. Now since R is transitive, and since aRb and bRa, it follows that aRa. Therefore R is reflexive.

Your answer to shows that this proof cannot be correct. What is the first incorrect statement in the proof, and why is it incorrect?