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In fairness, if you handed this in for homework in a sophomore algebra course, you likely wouldn't get credit (you've definitely proven the "only if"; the "if"
by stephencanon 14y ago
In fairness, if you handed this in for homework in a sophomore algebra course, you likely wouldn't get credit (you've definitely proven the "only if"; the "if" is a bit murky). However, it's not too much of a stretch to clean it up into a proper proof.
- dllthomas 14y agoIn what way did the aabb = a(ab)b = a(ba)b = abab fail to prove the if? edit: I did fail to re-state it, but figured it was obvious in the not-really-formal-proof setting. If that's all you meant by "a bit murky" then nevermind.
- stephencanon 14y agoYou have very clearly established the following: ab = ba --> (ab)^2 = a^2b^2 It is less obvious that you have proven that: (ab)^2 = a^2b^2 --> ab = ba If I were grading a sophomore algebra class, I would expect to see something along the lines of: Suppose (ab)^2 = a^2b^2. Re-associating gives us a(ba)b = a(ab)b; multiplying on the left and right by the inverses of a and b gives the result. In any domain outside of a sophomore algebra class, I happily accept much briefer and more hand-wavy proofs.
- dllthomas 14y agoAh, when I read your response I flipped the order of the problem around in recalling it, so had the if and only-if backwards. Yes, I was handwavy there but it seemed clear enough for the setting (which you seem to be granting anyway) - just wanted to be sure I wasn't misunderstanding something. Thanks :)