I am studying Velleman's "How to prove it: a structured approach" and I have to say that it is one of the best decision I have taken. Right now I am working on the exercises that are on Velleman's page along with the java applet "Proof Designer" and I have the feeling I start to get a bit how proofs work. But still... This proof is really making me think and I don't see how I can get out of it. Assume [itex]\wp[/itex](A)[itex]\cup[/itex][itex]\wp[/itex](B)=[itex]\wp[/itex](A[itex]\cup[/itex]B) Prove that A[itex]\subseteq[/itex]B or B[itex]\subseteq[/itex]A. [Suggested Exercise no.20] I have to admit I tried basically everything (i.e. contradiction, cases) but I don't really go anywhere close to the solution. For example, proof designer asks you to define a bounded variable if it is in the "given" section. I rephrase [itex]\wp[/itex](A)[itex]\cup[/itex][itex]\wp[/itex](B)=[itex]\wp[/itex](A[itex]\cup[/itex]B) and I don't know how to define the subset X of A and B (out of desperation I put X=A but doesn't look a great idea). Still I am not sure if it is a matter of not knowing how to properly use the software (which is actually quite easy) or I simply don't know how to work out the proof.